From 3713ae171224ab2440f8dab2104f9c9d4170f8db Mon Sep 17 00:00:00 2001 From: Chandu Bolisetti Date: Thu, 23 Aug 2018 18:24:07 -0600 Subject: [PATCH] Reorganizing theory and user manuals. Refs #156. --- doc/content/bib/mastodon.bib | 10 - doc/content/manuals/include/bcs/bcs-user.md | 62 ++ doc/content/manuals/include/bcs/drm-theory.md | 34 + .../manuals/include/bcs/intro_bcs-theory.md | 1 + .../include/bcs/non_reflecting-theory.md | 27 + .../include/bcs/preset_acceleration-theory.md | 10 + .../include/bcs/seismic_force-theory.md | 27 + .../include/contact/intro_contact-theory.md | 5 + .../include/contact/thin_layer-theory.md | 57 ++ .../damping/frequency_independent-theory.md | 3 + .../include/damping/intro_damping-theory.md | 10 + .../include/damping/rayleigh-theory.md | 108 +++ .../include/executioners/executioner-user.md | 15 + .../include/executioners/hht-theory.md | 9 + .../include/executioners/newmark-theory.md | 16 + .../executioners/time_integration-theory.md | 3 + .../fault_rupture/fault_rupture-theory.md | 60 ++ .../manuals/include/gravity/gravity-user.md | 22 + .../materials/intro_materials-theory.md | 22 + .../include/materials/intro_materials-user.md | 1 + .../manuals/include/materials/isoil-theory.md | 113 ++++ .../manuals/include/materials/isoil-user.md | 91 +++ .../materials/linearelasticsoil-theory.md | 8 + .../materials/linearelasticsoil-user.md | 10 + .../manuals/include/mesh/meshing-user.md | 59 ++ .../include/misc/getting_started-theory.md | 28 + .../include/misc/getting_started-user.md | 38 ++ .../include/misc/initial_stresses-user.md | 19 + .../model/governing_equations-theory.md | 12 + .../manuals/include/outputs/hsi-theory.md | 8 + .../include/outputs/intro_outputs-theory.md | 4 + .../include/outputs/intro_outputs-user.md | 20 + .../outputs/responsehistorybuilder-theory.md | 5 + .../responsespectracalculator-theory.md | 9 + .../manuals/include/ssi/intro_ssi-theory.md | 3 + .../include/ssi/soil_layering-theory.md | 25 + doc/content/manuals/theory/index.md | 633 +----------------- doc/content/manuals/user/index.md | 271 +------- doc/content/source/actions/ISoilAction.md | 8 +- doc/content/source/bcs/NonReflectingBC.md | 4 +- 40 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@article{housner1952hsi year={1952} } -@article{lsymer1969finite, - author={J.~Lsymer and R.~L.~Kuhlemeyer}, - title={Finite dynamic model for infinite media}, - journal={Journal of the Engineering Mechanics Division}, - volume={95}, - number={4}, - pages={859-878}, - year={1969} -} - @article{chiang1994anew, author={D.~Y.~Chiang and J.~L.~Beck}, title={A new class of distributed-element models for cyclic plasticity -I. Theory and application.}, diff --git a/doc/content/manuals/include/bcs/bcs-user.md b/doc/content/manuals/include/bcs/bcs-user.md new file mode 100644 index 0000000000..86287c133b --- /dev/null +++ b/doc/content/manuals/include/bcs/bcs-user.md @@ -0,0 +1,62 @@ +## Boundary Conditions + +Boundary conditions are required by MASTODON finite element analysis framework to be able to run +simple quasi-static and dynamic analysis. Herein, the basic boundary conditions are given and other +constraints that are used to solve for geotechnical earthquake engineering problems are presented +separately (e.g. periodic boundary conditions.) Following input block creates a fully fixed boundary condition at the bottom of a single element. + +!listing test/tests/materials/isoil/HYS_darendeli.i + start=BCs + end=Periodic + +In the above input, `type = PresetBC` sets the variable = disp_”related degree of freedom” with the +value = 0 which provides the fixity by defining a zero displacement at the node. boundary = 0 command +selects the nodes at the bottom surface of the element (labeled as surface 0) and assigns the +boundary conditions. + +#### Prescribed Displacement + +The preset displacement boundary condition can be used to apply a displacement time history to a +boundary (at the nodes). The displacement boundary condition first converts the user defined +displacement time history to an acceleration time history using Backward Euler finite difference +scheme. This acceleration is then integrated to get displacement using Newmark-beta method.  The +resulting displacement is then applied as a kinematic displacement boundary condition. The following +command can be used to apply the preset displacement boundary condition: + +!listing test/tests/materials/isoil/HYS_darendeli.i + start=top_x + end=Functions + +The above command should be embedded inside the BCs command block. “boundary = 5” assigns the preset +displacement to boundary 5 which, in this case, is a predefined boundary of a single element as +described in single element problem above. Alternatively, the boundary number can be identified using [Cubit](https://cubit.sandia.gov/) or [Trelis](https://www.csimsoft.com/trelis.jsp). “variable = disp_x” imposes the boundary condition on the x +direction. “beta” is the Newmark-beta integration parameter. The “function = top_disp” specifies the +function that defines the loading time history. It is defined in the “Functions” block as follows: + +!listing test/tests/materials/isoil/HYS_darendeli.i + start=Functions + end=Materials + +Displacement2.csv is the file, located in the same directory of the input file, containing the +displacement time history. The first column of this file should contain the time vector starting at +0.0. The second column should contain the displacement values. “type = PiecewiseLinear” defines the +type of the function which is in this case piecewise-linear. “format” specifies the format of the +data file, i.e. whether the data is in columns or rows. + +#### Prescribed Acceleration + +The preset acceleration boundary condition can be used to apply an acceleration time history to a +boundary. The preset acceleration boundary condition integrates the given acceleration time history +to get the displacement using Newmark-beta method. This displacement is then applied as a kinematic +displacement boundary condition. Syntax is the same as prescribing a displacement boundary condition +but with type = PresetAcceleration and the function describing time vs acceleration data instead of +time vs displacement. + +#### Periodic Boundary Conditions + +Periodic boundary conditions are used to constrain the nodes to move together in the specified +directions. The following input is an example applied on the single element problem above and should be embedded into the BCs block segment as: + +!listing test/tests/materials/isoil/HYS_darendeli.i + start=Periodic + end=top_x diff --git a/doc/content/manuals/include/bcs/drm-theory.md b/doc/content/manuals/include/bcs/drm-theory.md new file mode 100644 index 0000000000..f05b879e39 --- /dev/null +++ b/doc/content/manuals/include/bcs/drm-theory.md @@ -0,0 +1,34 @@ +#### Domain reduction method (DRM) + +Earthquake 'source-to-site' simulations require simulating a huge soil domain (order of many +kilometers) with a earthquake fault. The nuclear power plant structure, which is usually less than +100 m wide, is located very far from the earthquake fault, and the presence of the structure only +affects the response of the soil in the vicinity of the structure. In most of these situations, where +a localized feature such as a structure is present in a huge soil domain, the problem can be divided +into two parts: (i) a free-field 'source-to-site' simulation is run on the huge soil domain ( +[fig:DRM](a)) that does not contain the localized feature, and (ii) the forces from the free-field +simulation at one element layer, which is the element layer separating the bigger and smaller soil +domain, can be transferred to a much smaller domain containing the localized feature ( +[fig:DRM](b)). This method of reducing the domain is called the domain reduction method (DRM) +[citep:bielak2003domain]. [fig:DRM] is reproduced from [citet:bielak2003domain]. + +!media media/theory/DRM.png + style=width:100%;float:center; + id=fig:DRM + caption=Domain reduction method summary: (a) Big soil domain containing the earthquake fault + but not the localized feature. The displacements are obtained at the boundaries + $\Gamma$ and $\Gamma_e$ and are converted to equivalent forces. (b) Smaller soil + domain containing the localized feature but not the earthquake fault. The equivalent + forced calculated in (a) are applied at the boundaries $\Gamma$ and $\Gamma_e$. + +To convert the displacements at $\Gamma$ and $\Gamma_e$ from part (i) to equivalent forces, a finite +element model of the one element layer between $\Gamma$ and $\Gamma_e$ is simulated in two +steps. First, the boundary $\Gamma_e$ is fixed and the boundary $\Gamma$ is moved with the +displacements recorded at $\Gamma$. This step gives the equivalent forces at $\Gamma_e$. Second, the +boundary $\Gamma$ is fixed and the boundary $\Gamma_e$ is moved with the displacements recorded at +$\Gamma_e$. This steps gives the equivalent forces at $\Gamma$. + +Note: The meshes for the bigger soil domain and smaller soil domain need not align between $\Gamma$ +and $\Gamma_e$. The equivalent forces can be applied as point forces at the same coordinate location +at which nodes are present in the bigger model, irrespective of whether these locations correspond to +nodal locations in the smaller model. diff --git a/doc/content/manuals/include/bcs/intro_bcs-theory.md b/doc/content/manuals/include/bcs/intro_bcs-theory.md new file mode 100644 index 0000000000..a1897d4d27 --- /dev/null +++ b/doc/content/manuals/include/bcs/intro_bcs-theory.md @@ -0,0 +1 @@ +## Boundary Conditions diff --git a/doc/content/manuals/include/bcs/non_reflecting-theory.md b/doc/content/manuals/include/bcs/non_reflecting-theory.md new file mode 100644 index 0000000000..9fafd00501 --- /dev/null +++ b/doc/content/manuals/include/bcs/non_reflecting-theory.md @@ -0,0 +1,27 @@ +#### Non-reflecting boundary + +This boundary condition applies a Lysmer damper [citep:lysmer1969finite] on a given boundary to +absorb the waves hitting the boundary. To understand Lysmer dampers, let us consider an uniform +linear elastic soil column and say a 1D vertically propagating P wave is traveling through this soil +column. Then the normal stress at any location in the soil column is given by: + +\begin{equation} +\label{eqn:normal_stress} +\sigma = E \epsilon = E \frac{du}{dx} = \frac{E}{V_p} \frac{du}{dt}= \rho V_p \frac{du}{dt} +\end{equation} + +where, $E$ is the Young's modulus, $\sigma$ is the normal stress, $\epsilon$ is the normal strain, +$\rho$ is the density, $V_p = \sqrt{\frac{E}{\rho}}$ is the P-wave speed and $\frac{du}{dt}$ is the +particle velocity. Note that for a 3D problem, the P-wave speed is $V_p = \sqrt{\frac{E(1-\ +nu)}{(1+\nu)(1-2\nu)}}$. + +The stress in the above equation is directly proportional to the particle velocity which makes this +boundary condition analogous to a viscous damper with damping coefficient of $\rho V_p$. So +truncating the soil domain and placing this damper at the end of the domain is equivalent to +simulating wave propagation in an infinite soil column provided the soil is made of linear elastic +material and the wave is vertically incident on the boundary. + +If the soil is not linear elastic or if the wave is incident at an angle on the boundary, the waves +are not completely absorbed by the Lysmer damper. However, if the non-reflecting boundary is placed +sufficiently far from the region of interest, any reflected waves will get damped out by Rayleigh +damping or hysteretic material behavior before it reaches the region of interest. diff --git a/doc/content/manuals/include/bcs/preset_acceleration-theory.md b/doc/content/manuals/include/bcs/preset_acceleration-theory.md new file mode 100644 index 0000000000..2d494350e7 --- /dev/null +++ b/doc/content/manuals/include/bcs/preset_acceleration-theory.md @@ -0,0 +1,10 @@ +#### Preset acceleration + +If the ground excitation was measured at a depth within the soil by placing an accelerometer at that +location, then it is termed as a within-soil input. This time history contains the wave that was +generated by the earthquake (incoming wave) and the wave that is reflected off the free surface. This +ground excitation time history is usually available in the form of a acceleration time history. Since +MASTODON is a displacement controlled algorithm, i.e., displacements are the primary unknown +variables, the acceleration time history is first converted to the corresponding displacement time +history using Newmark time integration equation. This displacement time +history is then prescribed to the boundary. diff --git a/doc/content/manuals/include/bcs/seismic_force-theory.md b/doc/content/manuals/include/bcs/seismic_force-theory.md new file mode 100644 index 0000000000..a57833500f --- /dev/null +++ b/doc/content/manuals/include/bcs/seismic_force-theory.md @@ -0,0 +1,27 @@ +#### Seismic force + +In some cases, the ground excitation is measured at a rock outcrop (where rock is found at surface +level and there is no soil above it). To apply this to a location where rock is say $10$m deep and +there is soil above it, a sideset is created at $10$m depth and the ground excitation (converted into +a stress) is applied at this depth. To apply ground excitation as a stress, the input function should +be given as ground velocity. + +To convert a velocity applied normal to the boundary into a normal stress, the normal stress equation above can be used. Using a similar argument as discussed in the section above, to +convert a velocity applied tangential to the boundary into a shear stress, Equation +[eqn:shear_stress] can be used. + +\begin{equation} +\label{eqn:shear_stress} +\tau = \rho V_s \frac{du}{dt} +\end{equation} +where, $V_s$ is the shear wave speed and $\tau$ is the shear stress. + +In some situations, the ground motion measured at a depth within the soil is available. This ground +motion is the summation of the wave that enters and exits the soil domain. MASTODON has the +capability to extract the incoming wave from the within soil ground motion. To calculate the incoming +wave velocity, an iterative procedure is used. The initial guess for the incoming wave velocity +($v_i$) at time t is taken to be the same as the within soil velocity measured at that location. The +velocity at this boundary obtained from MASTODON ($v_{mastodon}$) is now going to be different from +the measured within soil velocity ($v_{measured}$) at time t. Half the difference between +$v_{mastodon}$ and $v_{measured}$ is added to $v_o$ and the iterations are continued until $v_i$ +converges (within a numerical tolerance). diff --git a/doc/content/manuals/include/contact/intro_contact-theory.md b/doc/content/manuals/include/contact/intro_contact-theory.md new file mode 100644 index 0000000000..7fe8868a92 --- /dev/null +++ b/doc/content/manuals/include/contact/intro_contact-theory.md @@ -0,0 +1,5 @@ +## Foundation-soil interface models + +The foundation-soil interface is an important aspect of NLSSI modeling. The foundation-soil interface +simulates geometric nonlinearities in the soil-structure system: gapping (opening and closing of gaps +between the soil and the foundation), sliding, and uplift. diff --git a/doc/content/manuals/include/contact/thin_layer-theory.md b/doc/content/manuals/include/contact/thin_layer-theory.md new file mode 100644 index 0000000000..c548d5e6e7 --- /dev/null +++ b/doc/content/manuals/include/contact/thin_layer-theory.md @@ -0,0 +1,57 @@ +#### Thin-layer method + +An efficient approach to modeling the foundation-soil interface is to create a thin layer of the +I-Soil material at the interface, as illustrated in [fig:thin_layer] below. + +!media media/theory/thin_layer.png + style=width:60%;margin-left:100px;float:center; + id=fig:thin_layer + caption=Modeling the foundation-soil interface as a thin layer for a sample surface foundation. + +The red layer between the foundation (green) and soil (yellow) is the thin layer of I-Soil. The +properties of this thin layer are then adjusted to simulate Coulomb friction between the +surfaces. The Coulomb-friction-type behavior can be achieved by modeling the material of the thin +soil layer as follows: + +1. Define an I-Soil material with a user-defined stress-strain curve. + +2. Calculate the shear strength of the thin layer as $\tau_{max}=\mu \sigma_N$ , where $\tau_{max}$ + is the shear strength, $\mu$ is the friction coefficient, and $\sigma_N$ is the normal stress on + the contact surface. The shear strength of the thin layer is the stress at which sliding starts at + the interface. Therefore, this shear strength should be proportional to the normal stress to + simulate Coulomb friction. This can be achieved by setting the initial shear strength equal to the + reference pressure, $p_{ref}$. The reference pressure can then be set to an arbitrary positive + value, such as the average normal stress at the interface from gravity loads. The shear strength + will then follow the equation + + \begin{equation} + \tau_{max} = \mu p_{ref} + \end{equation} + +3. Define the stress-strain curve to be almost elastic-perfectly-plastic, and such that the shear + modulus of the thin layer is equal to the shear modulus of the surrounding soil, in case of an + embedded foundation. If the foundation is resting on the surface such as in [fig:thin_layer] + above, the shear modulus of the thin layer soil should be as high as possible, such that the + linear horizontal foundation stiffness is not reduced due to the presence of the thin layer. A + sample stress-strain curve is shown in [fig:thin_layer_stress_strain] below. The sample curve in + the figure shows an almost bilinear shear behavior with gradual yielding and strain hardening, + both of which, are provided to reduce possible high-frequency response. High-frequency response is + likely to occur if a pure Coulomb friction model (elastic-perfectly-plastic shear behavior at the + interface) is employed, due to the sudden change in the interface shear stiffness to zero. + +!media media/theory/thin_layer_stress_strain.png + style=width:60%;margin-left:150px;float:center; + id=fig:thin_layer_stress_strain + caption=Sample shear-stress shear-strain curve for modeling the thin-layer interface using I-Soil. + +4. Turn on pressure dependency of the soil stress-strain curve and set $a_0$, $a_1$ and $a_2$ to 0, 0 + and 1, respectively. This ensures that the stress-strain curve scales linearly with the normal + pressure on the interface, thereby also increasing the shear strength in the above equation + linearly with the normal pressure, similar to Coulomb friction. + +5. Use a large value for the Poisson’s ratio, in order to avoid sudden changes in the volume of the + thin-layer elements after the yield point is reached. A suitable value for the Poisson’s ratio can + be calculated by trial and error. + +Following the above steps should enable the user to reasonably simulate geometric +nonlinearities. These steps will be automated in MASTODON in the near future. diff --git a/doc/content/manuals/include/damping/frequency_independent-theory.md b/doc/content/manuals/include/damping/frequency_independent-theory.md new file mode 100644 index 0000000000..051f6da4e5 --- /dev/null +++ b/doc/content/manuals/include/damping/frequency_independent-theory.md @@ -0,0 +1,3 @@ +#### Frequency-independent damping + +As seen in the previous sub-section, the damping ratio using Rayleigh damping varies with frequency. Although the parameters $\eta$ and $\zeta$ can be tuned to arrive at a constant damping ratio for a short frequency range, as the frequency range increases, the damping ratio no longer remains constant. For scenarios like these, where a constant damping ratio is required over a large frequency range, frequency independent damping formulations work better. This formulations is under consideration for adding to MASTODON. diff --git a/doc/content/manuals/include/damping/intro_damping-theory.md b/doc/content/manuals/include/damping/intro_damping-theory.md new file mode 100644 index 0000000000..a36bf8bb68 --- /dev/null +++ b/doc/content/manuals/include/damping/intro_damping-theory.md @@ -0,0 +1,10 @@ +## Damping + +When the soil-structure system (including both soil and concrete) +responds to an earthquake excitation, energy is dissipated in two primary +ways: (1)small-strain and hysteretic material damping, and (2) damping due to gapping, +sliding and uplift at the soil-foundation interface. Dissipation of +energy due to item (1) is modeled (approximately) using following methods: (i) viscous damping for small strain damping experienced at very small strain +levels ($\gamma$ $\leq 0.001 \%$) where the material behavior is largely linear viscoelastic; (ii) +hysteretic damping due to nonlinear hysteretic behavior of the material. +Dissipation of energy due to (2) is discussed in [foundation-soil interface models](#Foundation-soil interface models). This section discusses the damping that is present at small strain levels. diff --git a/doc/content/manuals/include/damping/rayleigh-theory.md b/doc/content/manuals/include/damping/rayleigh-theory.md new file mode 100644 index 0000000000..74ccaad76a --- /dev/null +++ b/doc/content/manuals/include/damping/rayleigh-theory.md @@ -0,0 +1,108 @@ +#### Rayleigh damping + +Rayleigh damping is the most common form of classical damping used in modeling structural dynamic problems. The more generalized form of classical damping, Caughey Damping [citep:caughey1960classical], is currently not implemented in MASTODON. Rayleigh damping is a specific form of Caughey damping that uses only the first two terms of the series. In this method, the viscous damping is proportional to the inertial contribution and contribution from the stiffness. This implies that in the matrix form of the governing equation, the damping matrix ($\mathbf{C}$) is assumed to be a linear combination of the mass ($\mathbf{M}$) and stiffness ($\mathbf{K}$) matrices, i.e., $\mathbf{C} = \eta \mathbf{M} +\zeta\mathbf{K}$. Here, $\eta$ and $\zeta$ are the mass and stiffness dependent Rayleigh damping parameters, respectively. + +The equation of motion (in the matrix form) in the presence of Rayleigh damping becomes: +\begin{equation} +\mathbf{M}\mathbf{\ddot{u}}+ (\eta \mathbf{M} + \zeta \mathbf{K})\mathbf{\dot{u}} +\mathbf{K}\mathbf{u} = \mathbf{F_{ext}} +\end{equation} + +The same equation of motion at any point in space and time (in the non-matrix form) is given by: +\begin{equation} +\rho\mathbf{\ddot{u}} + \eta \rho \mathbf{\dot{u}} + \zeta \nabla \cdot \frac{d}{dt}\sigma + \nabla \cdot \sigma = \mathbf{F_{ext}} +\end{equation} + +The degree of damping in the system depends on the coefficients $\zeta$ and $\eta$ as follows: +\begin{equation}\label{eqn:general_rayleigh} +\xi (f) = \frac{\eta}{2} \frac{1}{f} + \frac{\zeta}{2} f +\end{equation} + +where, $\xi(f)$ is the damping ratio of the system as a function of frequency $f$. The damping ratio as a function of frequency for $\zeta = 0.0035$ and $\eta = 0.09$ is presented in [fig:rayleigh]. + +!media media/theory/rayleigh.png + style=width:60%;margin-left:150px;float:center; + id=fig:rayleigh + caption=Damping ratio as a function of frequency. + + +#### Simulation of a constant damping ratio + +For the constant damping ratio scenario, the aim is to find $\zeta$ and $\eta$ such that the $\xi(f)$ is close to the target damping ratio $\xi_t$, which is a constant value, between the frequency range $[f_1, f_2]$. This can be achieved by minimizing the difference between $\xi_t$ and $\xi(f)$ for all the frequencies between $f_1$ and $f_2$, i.e., if + +\begin{equation} +I = \int_{f_1}^{f_2} \xi_t - \left(\frac{\eta}{2}\frac{1}{f} + \frac{\zeta}{2}f\right) df +\end{equation} + +Then, $\frac{dI}{d \eta} = 0$ and $\frac{dI}{d \zeta}=0$ results in two equations that are linear in $\eta$ and $\zeta$. Solving these two linear equations simultaneously gives: + +\begin{equation} +\begin{aligned} +\zeta &= \frac{\xi_t}{2 \pi} \; \frac{3}{(\Delta f)^2} \; \left(f_1 + f_2 - 2 \frac{f_1 f_1}{\Delta f} \; ln \frac{f_2}{f_1}\right) \\ +\eta &= 2 \pi \xi_t \; \frac{f_1 f_2}{\Delta_f} \; \left[ln \frac{f_2}{f_1}\; \left(2 + 6 \frac{f_1 f_2}{(\Delta_f)^2}\right) - \frac{3(f_1 + f_2)}{\Delta_f}\right] +\end{aligned} +\end{equation} + +##### Rayleigh damping for soils + +Small-strain material damping of soils is independent of loading frequency in frequency band of 0.01 Hz - 10 Hz ([cite:menq2003], [cite:shibuya2000damp],[cite:lopresti1997damp], and [cite:marmureanu2000damp]). The two mode Rayleigh damping is frequency dependent and can only achieve the specified damping at two frequencies while underestimating within and overestimating outside of these frequencies. The parameters $\eta$ and $\zeta$ for a given damping ratio can be calculated as follows: + +\begin{equation} + \begin{bmatrix} + \xi_i \\ + \xi_j + \end{bmatrix} + = + \frac{1}{4\pi} + \begin{bmatrix} + \frac {1}{f_i} & f_i \\ + \frac {1}{f_j} & f_j + \end{bmatrix} + \begin{bmatrix} + \eta \\ + \zeta + \end{bmatrix} +\end{equation} + +In case of two mode Rayleigh damping, [cite:kwok2007damp] suggests to use natural frequency and five times of it for the soil column of interest. In addition, selecting first mode frequency of soil column and higher frequency that corresponds to predominant period of the input ground motion is a common practice. + +Heterogeneities of the wave travel path may introduce scattering effect which leads to frequency dependent damping ([cite:campbell2009damp]). This type of damping is of the form ([cite:withers2015memory]): + +\begin{equation}\label{eqn:non_constant_damping} +\xi (f) = \begin{cases} + \xi_t, & \text{if}\ \; f \le f_T \\ + \xi_t \; \left(\frac{f_T}{f}\right)^\gamma, & \text{if}\ \;f > f_T + \end{cases} +\end{equation} + +where, $f_T$ is the frequency above which the damping ratio starts to +deviate from the constant target value of $\xi_t$, and $\gamma$ is +the exponent which lies between 0 and 1. Minimizing the difference +between [eqn:non_constant_damping] and +[eqn:general_rayleigh] with respect to $\eta$ and $\zeta$ for +all frequencies between $f_1$ and $f_2$ gives: + +\begin{equation} +\begin{aligned} +\zeta &= \frac{\xi_t}{2 \pi} \; \frac{6}{(\Delta f)^3} \; [b(f_1,f_2) - a(f_1, f_2) \; f_1 f_2] \\ +\eta &= 2 \pi \xi_t \; \frac{2 f_1 f_2}{(\Delta f)^3} \; [a(f_1, f_2)\; ({f_1}^2 + {f_2}^2 + f_1 f_2) - 3 b(f_1, f_2)] +\end{aligned} +\end{equation} + +where, the functions $a(f_1, f_2)$ and $b(f_1, f_2)$ are given by: + +\begin{equation} +\begin{aligned} +a(f_1, f_2) &= ln \frac{f_T}{f_1} + \frac{1}{\gamma} \; \left[1- \left(\frac{f_T}{f_2}\right)^\gamma\right] \\ +b(f_1, f_2) &= \frac{{f_T}^2 - {f_1}^2}{2} + \frac{{f_T}^\gamma}{2-\gamma} \; ({f_2}^{2-\gamma} - {f_T}^{2-\gamma}) +\end{aligned} +\end{equation} + +Also, $\xi_t$ for soils is inversely proportional to the shear wave +velocity ($V_s$) and a commonly used expression for $\xi_t$ of soil +is: + +\begin{equation} +\xi_t = \frac{5}{V_s} +\end{equation} + +where, $V_s$ is in m/s. diff --git a/doc/content/manuals/include/executioners/executioner-user.md b/doc/content/manuals/include/executioners/executioner-user.md new file mode 100644 index 0000000000..82c710a1cb --- /dev/null +++ b/doc/content/manuals/include/executioners/executioner-user.md @@ -0,0 +1,15 @@ +## Executioner + +The executioner input specifies the type of the analysis (the solver) and the corresponding parameters. There are two main types of executioners in MASTODON: steady and transient. Herein, an example is provided to run a transient analysis that will be used later on examples related to geotechnical earthquake engineering +applications. + +!listing test/tests/materials/isoil/HYS_darendeli.i + start=Executioner + end=Postprocessors + +The `type = Transient` command is self-explanatory and sets the executioner type to +transient. `solve_type = PJFNK` is the default option (preconditioned Jacobian Free Newton-Krylov) used +to solve the transient problems in MASTODON without constructing full stiffness matrix. `dt = 0.01` (is set to 0.01 sec here for demonstration) determines the initial time step and the user +is allowed to set a minimum time step, `dtmin`, for the analysis. `start_time` and `end_time` specifies +where the transient analysis will start and where it will end. Further information on available +options can be found in [/Executioner/index.md]. diff --git a/doc/content/manuals/include/executioners/hht-theory.md b/doc/content/manuals/include/executioners/hht-theory.md new file mode 100644 index 0000000000..c8c03b0f5b --- /dev/null +++ b/doc/content/manuals/include/executioners/hht-theory.md @@ -0,0 +1,9 @@ +#### Hilber-Hughes-Taylor (HHT) time integration + +The HHT time integration scheme [citep:hughes2000thefinite] is built upon Newmark time integration method to provide an unconditionally stable and second order accurate numerical scheme with the ability to damp out high frequency numerical noise. Here, in addition to the Newmark equations, the equation of motion is also altered resulting in: + +\begin{equation} +\rho\mathbf{\ddot{u}}(t+\Delta t) + \nabla \cdot [(1+\alpha) \sigma(t+\Delta t) - \alpha \sigma(t)] = \mathbf{F_{ext}}(t+\alpha\Delta t) +\end{equation} + +Here, $\alpha$ is the HHT parameter. The optimum parameter combination to use for this time integration scheme is $\beta = \frac{1}{4}(1-\alpha)^2$, $\gamma = \frac{1}{2} - \alpha$, and $-0.3 \le \alpha \le 0$. diff --git a/doc/content/manuals/include/executioners/newmark-theory.md b/doc/content/manuals/include/executioners/newmark-theory.md new file mode 100644 index 0000000000..1ccbb1bc6c --- /dev/null +++ b/doc/content/manuals/include/executioners/newmark-theory.md @@ -0,0 +1,16 @@ +#### Newmark time integration + +In Newmark time integration [citep:newmark1959amethod], the acceleration and velocity at $t+\Delta t$ are written in terms of the displacement ($\mathbf{u}$), velocity ($\mathbf{\dot{u}}$) and acceleration ($\mathbf{\ddot{u}}$) at time $t$ and the displacement at $t+\Delta t$. + +\begin{equation} \label{eqn:Newmark} +\begin{aligned} +\mathbf{\ddot{u}}(t+\Delta t) &= \frac{\mathbf{u}(t+\Delta t)-\mathbf{u}(t)}{\beta \Delta t^2} - \frac{\mathbf{\dot{u}}(t)}{\beta \Delta t}+\frac{\beta -0.5}{\beta}\mathbf{\ddot{u}}(t) \\ +\mathbf{\dot{u}}(t+ \Delta t) &= \mathbf{\dot{u}}(t)+ (1-\gamma)\Delta t \mathbf{\ddot{u}}(t) + \gamma \Delta t \mathbf{\ddot{u}}(t+\Delta t) +\end{aligned} +\end{equation} + +In the above equations, $\beta$ and $\gamma$ are Newmark time integration parameters. Substituting the above two equations into the equation of motion will result in a linear system of equations ($\mathbf{Au}(t+\Delta t) = \mathbf{b}$) from which $\mathbf{u}(t+\Delta t)$ can be estimated. + +For $\beta = 0.25$ and $\gamma = 0.5$, the Newmark time integration scheme is the same as the trapezoidal rule. The trapezoidal rule is an unconditionally stable integration scheme, i.e., the solution does not diverge for any choice of $\Delta t$, and the solution obtained from this scheme is second order accurate. One disadvantage with using trapezoidal rule is the absence of numerical damping to damp out any high frequency numerical noise that is generated due to the discretization of the equation of motion in time. + +The Newmark time integration scheme is unconditionally stable for $\gamma \ge \frac{1}{2}$ and $\beta \ge \frac{1}{4}\gamma$. For $\gamma > 0.5$, high frequency oscillations are damped out, but the solution accuracy decreases to first order. diff --git a/doc/content/manuals/include/executioners/time_integration-theory.md b/doc/content/manuals/include/executioners/time_integration-theory.md new file mode 100644 index 0000000000..7f4b0906a1 --- /dev/null +++ b/doc/content/manuals/include/executioners/time_integration-theory.md @@ -0,0 +1,3 @@ +## Time integration + +To solve the governing equation above for $\mathbf{u}$, an appropriate time integration scheme needs to be chosen. Newmark and Hilber-Hughes-Taylor (HHT) time integration schemes are two of the commonly used methods in solving wave propagation problems. diff --git a/doc/content/manuals/include/fault_rupture/fault_rupture-theory.md b/doc/content/manuals/include/fault_rupture/fault_rupture-theory.md new file mode 100644 index 0000000000..8d04e8c81a --- /dev/null +++ b/doc/content/manuals/include/fault_rupture/fault_rupture-theory.md @@ -0,0 +1,60 @@ +## Earthquake fault rupture + +The orientation of an earthquake fault is described using three directions - strike ($\phi_s$), dip +($\delta$) and slip direction ($\lambda$) as shown in [fig:fault_orientation], which is courtesy of +[citet:aki2012quantitative]. + +!media media/fault_orientation.png + style=width:80%;margin-left:100px; + id=fig:fault_orientation + caption=Definition of the fault-orientation parameters - strike $\phi_s$, dip $\delta$ and + slip direction $\lambda$. The slip direction is measured clockwise around from north, + with the fault dipping down to the right of the strike direction. Strike direction is + also measured from the north. $\delta$ is measured down from the horizontal. + +In MASTODON, earthquake fault is modeled using a set of point sources. The seismic moment ($M_o$) of +the earthquake point source in the fault specific coordinate system is: + +\begin{equation} +M_o(t) = \mu A \bar{u}(t) +\end{equation} + +where, $\mu$ is the shear modulus of the soil, $A$ is the area of fault rupture and $\bar{u}(t)$ is +the spatially averaged slip rate of the fault. + +When this seismic moment is converted into the global coordinate system (x, y and z) with the x +direction oriented along the geographic north and z direction along the soil depth, the resulting +moment can be written in a symmetric $3 \times 3$ matrix form whose components are as follows: + +\begin{equation} +\begin{aligned} +M_{xx}(t) &= -M_o(t)(\sin \delta \cos \lambda \sin2 \phi_s + \sin 2\delta \sin\lambda \sin^2 \phi_s) \\ +M_{xy}(t) &= M_{yx}(t) = M_o(t)(\sin\delta \cos \lambda \cos 2 \phi_s + \frac{1}{2} \sin 2\delta \sin \lambda \sin 2 \phi_s) \\ +M_{xz}(t) &= M_{zx}(t) = -M_o(t)(\cos \delta \cos \lambda \cos \phi_s + \cos 2\delta \sin \lambda \sin 2\phi_s) \\ +M_{yy}(t) &= M_o(t)(\sin \delta \cos \lambda \sin 2 \phi_s - \sin 2 \delta \sin \lambda \cos^2 \phi_s) \\ +M_{yz}(t) &= M_{zy}(t) = -M_o(t)(\cos \delta \cos \lambda \sin \phi_s - \cos 2\delta \sin\lambda \cos\phi_s) \\ +M_{zz}(t) &= M_o(t) \sin 2\delta \sin \lambda +\end{aligned} +\end{equation} + +Each component of the above matrix is a force couple with the first index representing the force +direction and the second index representing the direction in which the forces are separated (see +[fig:source_direction]; [citet:aki2012quantitative]). + +!media media/source_direction.png + style=width:60%;margin-left:150px; + id=fig:source_direction + caption=The nine different force couples required to model an earthquake source. + +The total force in global coordinate direction $i$ resulting from an earthquake source applied at +point $\vec{\zeta}$ in space is then: + +\begin{equation} +f_i(\vec{x}, t) = - \sum_{j=1}^{3} \frac{\partial M_{ij}(\vec{x}, t)}{\partial x_j} = \sum_{j=1}^{3} M_{ij}(t) \frac{\partial \delta (\vec{x} - \vec{\zeta})}{\partial x_j} +\end{equation} +where, $\delta(.)$ is the delta function in space. + +When many earthquake sources are placed on the earthquake fault, and they rupture at the same time +instant, then an approximation to a plane wave is generated. If one of the point sources is specified +as the epicenter and the rupture speed ($V_r$) is provided, then the other point sources start +rupturing at $d/V_r$, where $d$ is the distance between the epicenter and the other point source. diff --git a/doc/content/manuals/include/gravity/gravity-user.md b/doc/content/manuals/include/gravity/gravity-user.md new file mode 100644 index 0000000000..03ec0f2dd2 --- /dev/null +++ b/doc/content/manuals/include/gravity/gravity-user.md @@ -0,0 +1,22 @@ +## Gravity Loading + +Gravity loading can be applied on the domain to achieve the stress state due to gravity. MOOSE +includes built-in commands to apply the gravity as a body force throughout the transient +analysis. The following command can be used to apply the gravitational field on the modeled domain: + +!listing test/tests/materials/isoil/HYS_darendeli.i + start=gravity + end=AuxKernels + +The above command should be embedded inside the [Kernels](manuals/user/index.md) command +block. variable = disp_z activates the gravity in z direction and value = -9.81 is assigned to +specify the magnitude and direction of the gravitational field. + +Gravity command applies an external body force along the specified direction on the domain of +interest. In order to bring the system to equilibrium, a transient analysis with some viscous damping +is necessary. This approach causes fluctuations on the stress and strains at the beginning of the +analysis because of the elements being initially at zero stress state. Viscous damping removes the +fluctuations and brings the system to equilibrium. Once the system equilibrates, the gravity stresses +are obtained along with the displacements due to the gravity loading. “Initial Stress” command is +available in MASTODON framework to eliminate the need for a separate transient analysis. In addition, +no displacements, or strains due to gravity result. diff --git a/doc/content/manuals/include/materials/intro_materials-theory.md b/doc/content/manuals/include/materials/intro_materials-theory.md new file mode 100644 index 0000000000..9166f6954d --- /dev/null +++ b/doc/content/manuals/include/materials/intro_materials-theory.md @@ -0,0 +1,22 @@ +## Material models + +To model the mechanical behavior of a material, three components need to be defined at every point in space and time - strain, elasticity tensor, stress. + +1. +Strain+: Strain is a normalized measure of the deformation experienced by a material. In a 1-D + scenario, say a truss is stretched along its axis, the axial strain is the elongation of the truss + normalized by the length of the truss. In a 3D scenario, the strain is 3x3 tensor and there are + three different ways to calculate strains from displacements - small linearized total strain, + small linearized incremental strain, and finite incremental strain. Details about these methods + can be found in [modules/tensor_mechanics/index.md]. + +2. +Elasticity Tensor+: The elasticity tensor is a 4th order tensor with a maximum of 81 + independent constants. For MASTODON applications, the soil and structure are usually assumed to + behave isotropically, i.e., the material behaves the same in all directions. Under this + assumption, the number of independent elastic constants reduces from 81 to 2. The two independent + constants that are usually provided for the soil are the shear modulus and Poissons's ratio, and + for the structure it is the Young's modulus and Poisson's ratio. + +3. +Stress+: The stress at a point in space and time is a 3x3 tensor which is a function of the + strain at that location. The function that relates the stress tensor to the strain tensor is the + constitutive model. Depending on the constitutive model, the material can behave elastically or + plastically with an increment in strain. diff --git a/doc/content/manuals/include/materials/intro_materials-user.md b/doc/content/manuals/include/materials/intro_materials-user.md new file mode 100644 index 0000000000..db0344ad7e --- /dev/null +++ b/doc/content/manuals/include/materials/intro_materials-user.md @@ -0,0 +1 @@ +## Materials diff --git a/doc/content/manuals/include/materials/isoil-theory.md b/doc/content/manuals/include/materials/isoil-theory.md new file mode 100644 index 0000000000..2cd0909100 --- /dev/null +++ b/doc/content/manuals/include/materials/isoil-theory.md @@ -0,0 +1,113 @@ +#### Nonlinear hysteretic constitutive model for soils (I-soil) + +I-soil ([citet:numanoglu2017phd]) is a three dimensional, physically motivated, piecewise linearized +nonlinear hysteretic material model for soils. The model can be represented by shear type +parallel-series distributed nested components (springs and sliders) in one dimensional shear stress +space and its framework is analogous to the distributed element modeling concept developed by +[citet:iwan1967on]. The model behavior is obtained by superimposing the stress-strain response of +nested components. Three dimensional generalization follows [citet:chiang1994anew] and uses von Mises +(independent of effective mean stress) and/or Drucker-Prager (effective mean stress dependent) type +shear yield surfaces depending on user's choice. The yield surfaces are invariant in the stress space +[fig:yield_surface] . Thus, the model does not require kinematic hardening rule to model un/reloading +stress-strain response and preserves mathematical simplicity. + +!media media/theory/yield_surface.png + style=width:40%;margin-left:200px;float:center; + id=fig:yield_surface + caption=Invariant yield surfaces of the individual elastic-perfectly curves + (after Chiang and Beck, 1994). + +The current version of I-soil implemented in MASTODON utilizes Masing type un/reloading behavior and +is analogous to MAT79 (LS-DYNA) material model but does not exhibit numerical instability observed in +MAT79 ([citet:numanoglu2017conf]). Masing type un/reloading is inherently achieved by the model +because upon un/reloading the yielded nested components regain stiffness and strength. The cyclic +response obtained from current version of the model is presented in +[fig:1D_isoil_representation]. Reduction factor type modification on un/reloading behavior +([citet:phillips2009damping]; [citet:numanoglu2017nonmasing]) is an ongoing study within MASTODON +framework. + +!media media/theory/1D_isoil_representation.png + style=width:60%;margin-left:200px;float:center; + id=fig:1D_isoil_representation + caption=I-soil model details: (a) 1D representation by springs; (b) example monotonic and + cyclic behavior of four nested component model (reprinted from Baltaji et al., 2017). + +Main input for the current version of I-soil in MASTODON is a backbone curve at a given reference pressure. MASTODON provides variety of options to built backbone curve for a given soil type and reference pressure using following methods : + +1. User-defined backbone curve (soil\_type = 'user_defined'): The backbone curve can be provided in a + .csv file where the first column is shear strain points and the second column is shear stress + points. The number of nested components that will be generated from this backbone curve depends on + the number of discrete shear strain - shear stress pairs defined in the .csv file. When layered + soil profile is present, .csv file for each reference pressure can be provided to the + corresponding elements in the mesh. + +2. Darendeli backbone curve (soil\_type = 'darandeli'): The backbone curve can be auto-generated + based on empirical relations obtained from laboratory tests. [citet:darendeli2001development] + presents a functional form for normalized modulus reduction curves obtained from resonant column - + torsional shear test for variety of soils. MASTODON utilizes this study and auto-generates the + backbone shear stress - strain curves. The inputs for this option are (1) small strain shear + modulus, (2) bulk modulus, (3) plasticity index, (4) overconsolidation ratio, (5) reference + effective mean stress ($p_{ref}$) at which the backbone is constructed, and (6) number of shear + stress - shear strain points preferred by user to construct piecewise linear backbone curve. All + the other parameters except the number of shear stress - shear strain points can be provided as a + vector for each soil layer. + + Darendeli (2001) study extrapolates the normalized modulus reduction curves after 0.1% shear + strains. This extrapolation causes significant over/under estimation of the shear strength implied + at large strains for different type of soils at different reference effective mean stresses + [citep:hashash2010recent]. Thus user should be cautious about implied shear strength when + utilizing this option. + +3. General Quadratic/Hyperbolic (GQ/H) backbone curve (soil\_type = 'gqh'): + [citet:darendeli2001development] study presented in previous item constructs the shear stress - + shear strain curves based on experimentally obtained data. At small strains, the data is obtained + using resonant column test, and towards the moderate shear strain levels, torsional shear test + results are used. Large strain data are extrapolation of the small to medium shear strain + data. This extrapolation underestimates or overestimates the shear strength mobilized at large + shear strains. Therefore, implied shear strength correction is necessary to account for the + correct shear strength at large strains. GQ/H constitutive model proposed by + [citet:groholski2016simplified] has a unique curve fitting scheme embedded into the constitutive + model that accounts for mobilized strength at large shear strains by controlling the shear + strength. This model uses taumax, theta\_1 through 5, small strain shear modulus, bulk modulus and + number of shear stress - shear strain points preferred by user to construct piecewise linear + backbone curve. The parameter taumax is the maximum shear strength that can be mobilized by the + soil at large strains. The parameters theta\_1 through 5 are the curve fitting parameters and can + be obtained using DEEPSOIL [citep:hashash2016deepsoil]. Other than the number of points, all the + other parameters can be given as a vector for the different soil layers. The number of points, + which determines the number of elastic-perfectly plastic curves to be generated, is constant for + all soil layers. + +Once the backbone curve is provided to I-soil, the model determines the properties for nested +components presented in [fig:1D_isoil_representation]. The stress integration for each nested +component follows classic elastic predictor - plastic corrector type radial return algorithm +([citet:simo2006computein]) and model stress is obtained by summing the stresses from each nested +component: + +\begin{equation} +\tau = \sum_{k=1}^{i} G_{k} * \gamma + \sum_{k=i+1}^{n} {\tau_{y}}^k +\end{equation} + +where $\tau$ is the total shear stress, $G_{k}$ is the shear modulus of the $k^{th}$ nested +component, $\gamma$ is the shear strain, ${\tau_{y}}^k$ is the yield stress of the $k^{th}$ nested +component, and $i$ represents the number of components that have not yet yielded out of the $n$ total +nested components. + +Small strain shear modulus can be varied with effective mean stress via: + +\begin{equation} + G(p) = G_0 \left(\frac{p-p_0}{p_{ref}}\right)^{b_{exp}} +\end{equation} + +where, $G_0$ is the initial shear modulus, $p_0$ is the tension pressure cut off and $b_{exp}$ is +mean effective stress dependency parameter obtained from experiments. The shear modulus reduces to +zero for any mean effective stress lower than $p_0$ to model the failure of soil in tension. Note +that the mean effective stress is positive for compressive loading. Thus, $p_0$ should be inputted as +negative. + +Yield criteria of the material can also be varied with effective mean stress dependent behavior as: + +\begin{equation} +\tau_y(p) = \sqrt{\frac{a_0 + a_1 (p-p_0) + a_2 (p-p_0)^2}{a_0 + a_1 p_{ref} + a_2 {p_{ref}}^2}} \tau_y(p_{ref}) +\end{equation} + +where, $a_0$, $a_1$ and $a_2$ are parameters that define how the yield stress varies with pressure. diff --git a/doc/content/manuals/include/materials/isoil-user.md b/doc/content/manuals/include/materials/isoil-user.md new file mode 100644 index 0000000000..69050b5196 --- /dev/null +++ b/doc/content/manuals/include/materials/isoil-user.md @@ -0,0 +1,91 @@ +#### The I_Soil System + +The I-soil material model is a nonlinear hysteretic soil model that is based on the distributed +element models developed by [citet:iwan1967on] and [citet:chiang1994anew]. In 1-D, this model takes +the backbone stress-strain curve and divides it into a set of elastic-perfectly plastic curves. The +total stress then is the sum of the stresses from the individual elastic-perfectly plastic curves. + +The three dimensional generalization of this model is achieved using von-Mises failure criteria for +each elastic-perfectly plastic curve resulting in an invariant yield surfaces in three-dimensional +stress space like in [fig:yieldsurface] (after [citet:chiang1994anew]). + +!media media/yield_surface.png + style=width:40%;float:right;margin:20px;padding:20px; + id=fig:yieldsurface + caption=Invariant yield surfaces of the individual elastic-perfectly curves. + +The following options are available for an automatic creation of the backbone curve: + +1. User-defined backbone curve (soil_type = 'user_defined'): The backbone curve can be provided in a + csv file where the first column is shear strain and the second column is shear stress. The number + of elastic-perfectly plastic curves that will be generated from this backbone curve depends on the + number of entries in the data file. When many soil layers are present, a vector of data files can + be provided as input. The size of this vector should equal the number of soil layers. Also the + number of entries in each data file should be the same. + +2. Darendeli backbone curve (soil_type = 'darendeli'): The backbone curve can be auto-generated based + on empirical data for common soil types. [citet:darendeli2001development] presents a functional + form that can be used to create the backbone shear stress - strain curves based on the + experimental results obtained from resonant column and torsional shear tests. This functional form + requires the initial shear modulus, initial bulk modulus, plasticity index, over consolidation + ratio, reference mean confining pressure (p_ref) and number of points as input. Other than the + number of points, all the other parameters can be provided as a vector for each soil layer. The + number of points, which determines the number of elastic-perfectly plastic curves to be generated, + is constant for all soil layers. + +3. General Quadratic/Hyperbolic (GQ/H) backbone curve (soil_type = 'gqh'): + [citet:darendeli2001development] study constructs the shear stress-strain curves based on + experimentally obtained data. At small strains the data is obtained using resonant column test, + and towards the medium shear strain levels the torsional shear test results are used. The values + are extrapolated at the large strain levels. This extrapolation may underestimate or overestimate + the shear strength at large strains. Therefore, shear strength correction is necessary to account + for the correct shear strength at large strains. GQ/H model proposed by + [citet:groholski2016simplified] has a curve fitting scheme that automatically corrects the + reference curves provided by [citet:darendeli2001development] based on the specific shear strength + at the large strains. This model requires taumax, theta_1 through 5, initial shear modulus, + initial bulk modulus and number of points as input. The parameter taumax is the maximum shear + stress that can be generated in the soil. The parameters theta_1 through 5 are the curve fitting + parameters and can be obtained using DEEPSOIL [citep:hashash2016deepsoil]. Other than the number + of points, all the other parameters can be given as a vector for the different soil layers. The + number of points, which determines the number of elastic-perfectly plastic curves to be generated, + is constant for all soil layers. + +4. Thin-layer friction backbone curve (soil_type = 'thin_layer'): This backbone curve is should be used for + the simulation of the foundation-soil interfaces using a formulation similar to Coulomb + friction. This option should be used for a thin layer of I-soil elements along the foundation-soil + interface. Using the 'thin_layer' option creates a bilinear backbone curve with a pre-yield shear + modulus equal to the the initial_shear_modulus, and a post-yield shear modulus equal to + initial_shear_modulus * hardening_ratio. The shear stress at yield is set to be equal to + friction_coefficient * instantaneous pressure. Using the 'thin_layer' option for the I-Soil + backbone curve automatically sets the values pressure_dependency = 'true', $a_0 = 0$, $a_1 = 0$ + and $a_2 = 1$. This ensures that the shear strength dependence is turned on, and the strength of + the thin layer increases linearly with instantaneous pressure. Note that since the strength is + directly proportional to pressure (and not normal stress), this is not identical to Coulomb + formulation. + +All the above backbone curves provide the behavior of the soil at a reference confining pressure +($p_{ref}$). When the confining pressure of the soil changes, the soil behavior also changes (if +pressure dependency is turned on). The shear modulus ($G(p)$) at a pressure $p$ is given by: + +\begin{equation} +G(p) = G_0 (\frac{p-p_0}{p_{ref}})^{b_{exp}}, +\end{equation} + +where $G_0$ is the initial shear modulus, $p_0$ is the tension pressure cut off and $b_{exp}$ is a +parameter obtained from experiments. The shear modulus reduces to zero for any pressure lower than +$p_0$ to model the failure of soil in tension. Note that compressive pressure is taken to be +positive. + +Similarly, the yield stress ($\sigma_y(p)$) of the elastic-perfectly plastic curve also changes when +the confining pressure changes. The yield stress ($\sigma_y(p)$) at a pressure $p$ is given as: + +\begin{equation} +\sigma_y(p) = \sqrt{\frac{a_0 + a_1 (p-p_0) + a_2 (p-p_0)^2}{a_0 + a_1 p_{ref} + a_2 {p_{ref}}^2}} \sigma_y(p_{ref}) , +\end{equation} + +where $a_0$, $a_1$ and $a_2$ are parameters obtained from experiments. + +To include pressure-dependent stiffness and yield strength calculation, pressure_dependency should be +set to true and b_exp, a_0, a_1, a_2, tension_pressure_cut_off and p_ref need to be provided as +input. Other than p_ref, all the other parameters are the same for all the soil layers. p_ref can be +provided as vector with information about each soil layer. diff --git a/doc/content/manuals/include/materials/linearelasticsoil-theory.md b/doc/content/manuals/include/materials/linearelasticsoil-theory.md new file mode 100644 index 0000000000..1afe024cf0 --- /dev/null +++ b/doc/content/manuals/include/materials/linearelasticsoil-theory.md @@ -0,0 +1,8 @@ +#### Linear elastic constitutive model + +In scenarios where the material exhibits a linear relation between stress and strain, and does not +retain any residual strain after unloading, is called a linear elastic material. In linear +elasticity, the stress tensor ($\sigma$) is calculated as $\sigma = \mathcal{C}\epsilon$, where +$\mathcal{C}$ is the elasticity tensor, and $\epsilon$ is the strain tensor. This material model is +currently used for numerically modeling the behavior of concrete and other materials used for +designing a structure in MASTODON. diff --git a/doc/content/manuals/include/materials/linearelasticsoil-user.md b/doc/content/manuals/include/materials/linearelasticsoil-user.md new file mode 100644 index 0000000000..db28376e64 --- /dev/null +++ b/doc/content/manuals/include/materials/linearelasticsoil-user.md @@ -0,0 +1,10 @@ +#### Linear elastic soil + +A linear elastic soil can be defined in MASTODON by defining a elasticity tensor using +ComputeIsotropicElasticityTensor, stress calculator using ComputeLinearElasticStress, and a small +strain calculator using ComputeSmallStrain. Density can be defined using the +GenericConstantMaterial. An example of these input blocks is shown below: + +!listing test/tests/dirackernels/seismic_source/one_seismic_source.i + start=Materials + end=Executioner diff --git a/doc/content/manuals/include/mesh/meshing-user.md b/doc/content/manuals/include/mesh/meshing-user.md new file mode 100644 index 0000000000..5522b2a957 --- /dev/null +++ b/doc/content/manuals/include/mesh/meshing-user.md @@ -0,0 +1,59 @@ +## Creating a mesh + +The first thing necessary to run a finite element analysis is a mesh. Generating a mesh of interest in MASTODON can be achieved using the inbuilt [mesh generator](source/mesh/GeneratedMesh.md) or by importing an existing mesh. Imported meshes should be of the exodus format and can be generated using the meshing software, [Cubit](https://cubit.sandia.gov/). Cubit is freely available for users in the U.S. federal government and national labs. For other users, the commercial version of Cutbit called [Trelis](https://www.csimsoft.com/trelis.jsp) is recommended. Usage of the inbuilt mesh generator for a simple problem is demonstrated below. + +For any analysis, the number +of dimensions and the degrees of freedom should be specified in the beginning of the `Mesh` block in the input file. An +example input is provided below: + +!listing test/tests/materials/isoil/HYS_darendeli.i + start=Mesh + end=GlobalParams + +This input creates a single brick element using the MASTODON mesh generator with 8 nodes, 8 gauss +quadrature points, and unit length in each edge. `dim = 3` specifies that the domain is three-dimensional, and nx, ny, and nz specify the number of elements in X, Y and Z directions, respectively. + +[fig:single_element] presents a generic three-dimensional brick element along with node and surface +labels. Generated brick element is automatically assigned a block number (block 0 in this case) and +each side of the brick is automatically assigned a surface number. + +!media media/user/single_element.png + style=width:60%;margin-left:150px;float:center; + id=fig:single_element + caption=Single brick element (a) node labels (b) surface labels. + +The next step is activating the global parameters. These are input parameters, which have the same +value across the input file. The global parameters are activated providing the following commands: + +!listing test/tests/materials/isoil/HYS_darendeli.i + start=GlobalParams + end=Variables + +By providing the above commands, the user specifies that if any object in the input file has a +parameter called “displacements”, that parameter would be set to “disp_x disp_y disp_z”, which are +the global displacement degrees of freedoms at the nodal points in x, y, and z directions, +respectively. + +Before or after (the sequence does not matter in the input file) the global parameters, the user has +to specify the solution variables using the following commands: + +!listing test/tests/materials/isoil/HYS_darendeli.i + start=Variables end=AuxVariables + +MASTODON allows user to define auxiliary variables. Auxiliary variables are not solved for by the +system but they are calculated from the solution variables. This is particularly necessary for +dynamic analysis, since the nodal accelerations and velocities are defined in this section. In +addition, the stresses and strains are defined as auxiliary variables to access the data later on to +inspect the results. Following commands are the examples of defining the auxiliary variables: + +!listing test/tests/materials/isoil/HYS_darendeli.i + start=AuxVariables + end=Kernels + +Using the above commands, user defines velocity, acceleration, stress, and strain auxiliary variables +to be used later on to ask for an output. order = CONSTANT and family=MONOMIAL calculates the average +stress within the element. + +The above command blocks forms the basis for all quasi-static and dynamic analyses that can be +conducted using MASTODON framework. Next section describes the Kernels related to physics involved in +the particular analysis. diff --git a/doc/content/manuals/include/misc/getting_started-theory.md b/doc/content/manuals/include/misc/getting_started-theory.md new file mode 100644 index 0000000000..11a3021390 --- /dev/null +++ b/doc/content/manuals/include/misc/getting_started-theory.md @@ -0,0 +1,28 @@ +## Introduction + +Multi-hazard Analysis for STOchastic time-DOmaiN phenomena (MASTODON) is +a finite element application that aims at analyzing the response of 3-D +soil-structure systems to natural and man-made hazards such as +earthquakes, floods and fire. MASTODON currently focuses on the +simulation of seismic events and has the capability to perform extensive +'source-to-site' simulations including earthquake fault rupture, +nonlinear wave propagation and nonlinear soil-structure interaction +(NLSSI) analysis. MASTODON is being developed to be a dynamic +probabilistic risk assessment framework that enables analysts to not +only perform deterministic analyses, but also easily perform +probabilistic or stochastic simulations for the purpose of risk +assessment. + +MASTODON is a MOOSE -based application and performs finite-element +analysis of the dynamics of solids, mechanics of interfaces and porous +media flow. It is equipped with effective stress space nonlinear hysteretic soil constitutive model (I-soil), and a u-p-U formulation to couple solid and fluid, as well as structural materials such as reinforced concrete. It includes interface models that +simulate gapping, sliding and uplift at the interfaces of solid media +such as the foundation-soil interface of structures. Absorbing boundary models for the simulation of infinite or +semi-infinite domains, fault rupture model for seismic source +simulation, and the domain reduction method for the input of complex, +three-dimensional wave fields are incorporated. Since MASTODON is a MOOSE -based +application, it can efficiently solve problems using standard +workstations or very large high-performance computers. + +This document describes the theoretical and numerical foundations of +MASTODON. diff --git a/doc/content/manuals/include/misc/getting_started-user.md b/doc/content/manuals/include/misc/getting_started-user.md new file mode 100644 index 0000000000..38993b7d55 --- /dev/null +++ b/doc/content/manuals/include/misc/getting_started-user.md @@ -0,0 +1,38 @@ +## Getting Started + +This section provides step-by-step instructions to define the basic components of a numerical +model for finite element analysis using the MASTODON framework. + + + +#### Kernels + +Kernels are related to the physics involved in the particular analysis. This document focusses on +kernel commands related to the geotechnical earthquake engineering applications. Further information +can be found in [/Kernels/index.md]. + +The main kernel that is used for quasi-static and dynamic analyses is +[modules/tensor_mechanics/index.md]. This kernel is used to solve the equation of motion without the +inertial effects. It requires information about the unknowns that are solved for. The following chunk +of commands can be used to activate dynamic tensor mechanics kernel along with Newmark-beta +integration scheme, inertial effects, and two mode Rayleigh viscous damping (both stiffness and mass +proportional damping): + +!listing test/tests/materials/isoil/HYS_darendeli.i + start=Kernels + end=BCs + +displacements = 'disp_x disp_y disp_z' line is not necessary if the displacements are already defined +as global parameters but are provided here for the sake of completeness. Beta and gamma are the +parameters of Newmark-beta integration scheme. Zeta and eta are the stiffness and mass matrix +coefficients of Rayleigh damping formulation (see [modules/tensor_mechanics/index.md] and +[Theory Manual](manuals/theory/index.md) for more information). Auxiliary kernels are specified to +calculate the acceleration and velocities using Newmark-beta scheme at the end of each time step +where the displacement is already solved and known. Lastly, stress_xy is defined as an auxiliary +variable. This is achieved by specifying the type of the Auxkernel as RankTwoAux. RankTwoAux means +that the source of the auxiliary variable is a rank two tensor, and the type of rank two tensor is +explicitly defined as stress tensor using the command “rank_two_tensor = stress”. Since, the variable +is stress_xy, the location in the stress tensor corresponding to stress_xy needs to be specified +using index_i (row index) and index_j (column index). To request for stress_xy, index_i is set to 0 +and index_j is set to 1. The next section explains the boundary conditions that are required to run a +simple, quasi-static and dynamic analyses. diff --git a/doc/content/manuals/include/misc/initial_stresses-user.md b/doc/content/manuals/include/misc/initial_stresses-user.md new file mode 100644 index 0000000000..d27f4f3c42 --- /dev/null +++ b/doc/content/manuals/include/misc/initial_stresses-user.md @@ -0,0 +1,19 @@ +## Initial Stresses + +Initial stress command is available to apply the prescribed initial stresses to the finite element +domain. If the user applies initial stress state that is equal to that of gravity loadings, upon +transient analysis, no vibration occurs and the domain reaches equilibrium with zero strains. This +eliminates the necessity of conducting separate analysis for free-field problems and allows the user +to start directly with a dynamic analysis such as base shaking etc. The following command can be used +within the material definition to activate the initial stresses in the material block: + +``` +initial_stress = '-4.204286 0 0 0 -4.204286 0 0 0 -9.810' +``` + +The nine numerical values fill the stress tensor in the following order - xx, xy, xz, yz, yy, yz, zx, +zy, and zz. The above example command activates -4.2025 units of stress along both xx and yy +direction, and -9.81 units of stress along zz direction. These stresses are the gravity stresses at +the mid-depth of the element in [fig:single_element] with density = 2. Therefore, the element will +immediately start with gravity stress conditions without any elastic deformations or stress +fluctuations. diff --git a/doc/content/manuals/include/model/governing_equations-theory.md b/doc/content/manuals/include/model/governing_equations-theory.md new file mode 100644 index 0000000000..74024721f2 --- /dev/null +++ b/doc/content/manuals/include/model/governing_equations-theory.md @@ -0,0 +1,12 @@ +## Governing equations + +The basic equation that MASTODON solves is the nonlinear wave equation: + +\begin{equation} +\label{eqn:governing_equation} + \rho \mathbf{\ddot{u}} + \nabla \cdot \sigma = \mathbf{F_{ext}} +\end{equation} + +where, $\rho$ is the density of the soil or structure that can vary with space, $\sigma$ is the stress at any point in space and time, $\mathbf{F_{ext}}$ is the external force acting on the system that can be in the form of localized seismic sources or global body forces such as gravity, and $\mathbf{\ddot{u}}$ is the acceleration at any point within the soil-structure domain. The left side of the equation contains the internal forces acting on the system with first term being the contribution from the inertia, and the second term being the contribution from the stiffness of the system. Additional terms would be added to this equation when damping is present in the system. The material stress response ($\sigma$) is described by the constitutive model, where the stress is determined as a function of the strain ($\epsilon$), i.e. $\sigma(\epsilon)$. Details about the material constitutive models available in MASTODON are presented in the section about [material models](#material-models). + +The above equation is incomplete and ill-conditioned without the corresponding boundary conditions. There are two main types of boundary conditions: (i) Dirichlet boundary condition which is a kinematic boundary condition where the displacement, velocity, or acceleration at that boundary is specified; (ii) Neumann boundary condition where a force or traction is applied at the boundary. All the special boundary conditions such as absorbing boundary condition are specialized versions of these broad boundary condition types. diff --git a/doc/content/manuals/include/outputs/hsi-theory.md b/doc/content/manuals/include/outputs/hsi-theory.md new file mode 100644 index 0000000000..a87673f9bd --- /dev/null +++ b/doc/content/manuals/include/outputs/hsi-theory.md @@ -0,0 +1,8 @@ +#### Housner spectrum intensity + +The response spectra is very useful for understanding the response of the system at one +location. However, if the response at multiple locations have to be compared, a single value that can +summarize the response at a location is much more useful. Housner spectrum intensity is the integral +of the velocity response spectra between 0.25-2.5 s (or 0.4-4 Hz). This packs the information from +the velocity response spectra at multiple frequencies into a single value and reasonably represents +the response at a location. diff --git a/doc/content/manuals/include/outputs/intro_outputs-theory.md b/doc/content/manuals/include/outputs/intro_outputs-theory.md new file mode 100644 index 0000000000..9944ed2ac6 --- /dev/null +++ b/doc/content/manuals/include/outputs/intro_outputs-theory.md @@ -0,0 +1,4 @@ +## Outputs + +This section presents some of the common post-processing tools available in MASTODON that help in +understanding the wave propagation, and response of structures and soils to earthquake excitation. diff --git a/doc/content/manuals/include/outputs/intro_outputs-user.md b/doc/content/manuals/include/outputs/intro_outputs-user.md new file mode 100644 index 0000000000..5906d4043f --- /dev/null +++ b/doc/content/manuals/include/outputs/intro_outputs-user.md @@ -0,0 +1,20 @@ +## Outputs + +The analysis results can be dumped to a csv file using postprocessor and output commands. The first +command should define the type of results that will be dumped into the csv file. The following +command is an example to specify the results to be outputted: + +!listing test/tests/materials/isoil/HYS_darendeli.i + start=Postprocessors + +Since, the stresses, accelerations, and velocities are defined as an auxiliary variable previously +([Kernels](manuals/user/index.md)), they can be directly called in this section. elementid = 0 +specifies the element number from which stress values will be pulled out. Note that MASTODON assigns +numbers to elements and nodes different than those of the input file provided by the user (for +user-defined mesh). To obtain the correct element and nodes numbers, the software Blot installed with +MOOSE or Paraview should be used. type = ElementVariableValue returns the average scalar stress value +computed within the specified block. Finally, the results can be dumped into a csv file by using +"Outputs command". Once the analysis is completed, the output csv file will be stored in the same +folder as the input file. The first column contains the “Time”, and the rest of the columns contain +the values asked in the "Postprocessors" command. Another form of output is the exodus file which can +be read by Blot. The user also has the option to request results in both csv and exodus formats. diff --git a/doc/content/manuals/include/outputs/responsehistorybuilder-theory.md b/doc/content/manuals/include/outputs/responsehistorybuilder-theory.md new file mode 100644 index 0000000000..392a4c0f47 --- /dev/null +++ b/doc/content/manuals/include/outputs/responsehistorybuilder-theory.md @@ -0,0 +1,5 @@ +#### Response histories + +In MASTODON, the time history of any nodal variable (displacement, velocity, acceleration) or +elemental variable (stress, strain) can be requested as output. The nodal variable time histories can +be requested at a set of nodes and this can help in visualizing the wave propagation. diff --git a/doc/content/manuals/include/outputs/responsespectracalculator-theory.md b/doc/content/manuals/include/outputs/responsespectracalculator-theory.md new file mode 100644 index 0000000000..e7dc987315 --- /dev/null +++ b/doc/content/manuals/include/outputs/responsespectracalculator-theory.md @@ -0,0 +1,9 @@ +#### Response spectra + +An important quantity that is used in understanding the response of a structure is the +velocity/acceleration response spectra. This contains information about the frequency content of the +velocity/acceleration at a particular location. The velocity/acceleration response spectra at a +frequency $\omega$ is obtained by exciting a single degree of freedom (SDOF) system with natural +frequency of $\omega$ with the velocity/acceleration time history recorded at that location, and +obtaining the peak velocity/acceleration experienced by the SDOF. This exercise is repeated for +multiple frequencies to obtain the full response spectra. diff --git a/doc/content/manuals/include/ssi/intro_ssi-theory.md b/doc/content/manuals/include/ssi/intro_ssi-theory.md new file mode 100644 index 0000000000..2e7ffc3e11 --- /dev/null +++ b/doc/content/manuals/include/ssi/intro_ssi-theory.md @@ -0,0 +1,3 @@ +## Site response and SSI + +MASTODON provides tools that simplifies site-response and SSI modeling. Currently, these tools include automated soil layering, and mesh sizing based on the maximum frequency of wave propagation. diff --git a/doc/content/manuals/include/ssi/soil_layering-theory.md b/doc/content/manuals/include/ssi/soil_layering-theory.md new file mode 100644 index 0000000000..5517e49d03 --- /dev/null +++ b/doc/content/manuals/include/ssi/soil_layering-theory.md @@ -0,0 +1,25 @@ +#### Soil layers and meshing + +Small strain properties (shear wave velocity, small strain modulus etc.) as well as mobilized shear strength of soils change with depth. Thus, in numerical models, soil profile (layers) is constructed to incorporate the depth dependent properties. The ground surface as well as layers that define the soil domain can be horizontal +or non-horizontal. For the horizontal ground surface and layering scenario, the location +of the interfaces can be provided as input and MASTODON will use that +information to generate a set of soil layers, each with a unique +identification number. These layer ids are later used to assign material +properties to each soil layer. The same procedure can also be used for +non-horizontal but planar soil layers by specifying the normal to the +plane and the interface locations measured along the normal direction. + +For scenarios where the soil layers are non-horizontal and non-planar, images (.jpg, .png, etc.) of the soil profile can be provided as input. The different soil layers are distinguished from the image by reading either the red, green or blue color value (as per user's directions) at each pixel. Gray scale images in which the red, green and blue values are all the same also work well for this purpose. For creating 3D soil layers, multiple 2D images with soil profiles at different 2-D cross-sections of the soil domain can be provided as input. + +Once the soil layers have been distinguished, it is necessary to ensure that the different soil layers are meshed such that they can accurately transmit waves of the required frequency. The optimum element size for each soil layer depends on the type of element used for meshing, cut-off frequency (f) of the wave and the shear wave velocity ($V_s$) of the soil layer. A minimum of 10 points is required per wavelength of the wave to accurately represent the wave in space [citep:coleman2016time]. The minimum wavelength ($\lambda_{min}$) is calculated as: + +\begin{equation} +\lambda_{min} = \frac{V_s}{f} +\end{equation} + +If linear elements such as QUAD4 or HEX8 are used, then the optimum mesh size is $\lambda_{min}/10$. If quadratic elements such as QUAD9 or HEX27 are used, then the optimum mesh size is $\lambda_{min}/5$. Using the minimum element size information, MASTODON refines the mesh such that the element size criterion is met and at the same time the layers separations are visible. An example of this meshing scheme is presented in [fig:adaptive_meshing] where a 2D soil domain is divided into 3 soil layers and these soil layers are meshed such that the element size criterion is satisfied. A denser mesh is created at the interface between different soil layers. + +!media media/theory/adaptive_mesh.png + style=width:60%;margin-left:150px;float:center; + id=fig:adaptive_meshing + caption=Auto-generated mesh for a soil domain with three non-horizontal non-planar soil layers. diff --git a/doc/content/manuals/theory/index.md b/doc/content/manuals/theory/index.md index a9575ec772..078643b8f9 100644 --- a/doc/content/manuals/theory/index.md +++ b/doc/content/manuals/theory/index.md @@ -1,636 +1,67 @@ # Theory manual -## Introduction + -Multi-hazard Analysis for STOchastic time-DOmaiN phenomena (MASTODON) is -a finite element application that aims at analyzing the response of 3-D -soil-structure systems to natural and man-made hazards such as -earthquakes, floods and fire. MASTODON currently focuses on the -simulation of seismic events and has the capability to perform extensive -'source-to-site' simulations including earthquake fault rupture, -nonlinear wave propagation and nonlinear soil-structure interaction -(NLSSI) analysis. MASTODON is being developed to be a dynamic -probabilistic risk assessment framework that enables analysts to not -only perform deterministic analyses, but also easily perform -probabilistic or stochastic simulations for the purpose of risk -assessment. +!include include/misc/getting_started-theory.md -MASTODON is a MOOSE -based application and performs finite-element -analysis of the dynamics of solids, mechanics of interfaces and porous -media flow. It is equipped with effective stress space nonlinear hysteretic soil constitutive model (I-soil), and a u-p-U formulation to couple solid and fluid, as well as structural materials such as reinforced concrete. It includes interface models that -simulate gapping, sliding and uplift at the interfaces of solid media -such as the foundation-soil interface of structures. Absorbing boundary models for the simulation of infinite or -semi-infinite domains, fault rupture model for seismic source -simulation, and the domain reduction method for the input of complex, -three-dimensional wave fields are incorporated. Since MASTODON is a MOOSE -based -application, it can efficiently solve problems using standard -workstations or very large high-performance computers. +!include include/model/governing_equations-theory.md -This document describes the theoretical and numerical foundations of -MASTODON. + -## Governing equations +!include include/executioners/time_integration-theory.md -The basic equation that MASTODON solves is the nonlinear wave equation: +!include include/executioners/newmark-theory.md -\begin{equation} -\label{eqn:governing_equation} - \rho \mathbf{\ddot{u}} + \nabla \cdot \sigma = \mathbf{F_{ext}} -\end{equation} + -where, $\rho$ is the density of the soil or structure that can vary with space, $\sigma$ is the stress at any point in space and time, $\mathbf{F_{ext}}$ is the external force acting on the system that can be in the form of localized seismic sources or global body forces such as gravity, and $\mathbf{\ddot{u}}$ is the acceleration at any point within the soil-structure domain. The left side of the equation contains the internal forces acting on the system with first term being the contribution from the inertia, and the second term being the contribution from the stiffness of the system. Additional terms would be added to this equation when damping is present in the system. The material stress response ($\sigma$) is described by the constitutive model, where the stress is determined as a function of the strain ($\epsilon$), i.e. $\sigma(\epsilon)$. Details about the material constitutive models available in MASTODON are presented in the section about [material models](#material-models). +!include include/damping/intro_damping-theory.md -The above equation is incomplete and ill-conditioned without the corresponding boundary conditions. There are two main types of boundary conditions: (i) Dirichlet boundary condition which is a kinematic boundary condition where the displacement, velocity, or acceleration at that boundary is specified; (ii) Neumann boundary condition where a force or traction is applied at the boundary. All the special boundary conditions such as absorbing boundary condition are specialized versions of these broad boundary condition types. +!include include/damping/rayleigh-theory.md -## Time integration +!include include/damping/frequency_independent-theory.md -To solve Equation [eqn:governing_equation] for $\mathbf{u}$, an appropriate time integration scheme needs to be chosen. Newmark and Hilber-Hughes-Taylor (HHT) time integration schemes are two of the commonly used methods in solving wave propagation problems. + -### Newmark time integration +!include include/ssi/intro_ssi-theory.md -In Newmark time integration [citep:newmark1959amethod], the acceleration and velocity at $t+\Delta t$ are written in terms of the displacement ($\mathbf{u}$), velocity ($\mathbf{\dot{u}}$) and acceleration ($\mathbf{\ddot{u}}$) at time $t$ and the displacement at $t+\Delta t$. +!include include/ssi/soil_layering-theory.md -\begin{equation} \label{eqn:Newmark} -\begin{aligned} -\mathbf{\ddot{u}}(t+\Delta t) &= \frac{\mathbf{u}(t+\Delta t)-\mathbf{u}(t)}{\beta \Delta t^2} - \frac{\mathbf{\dot{u}}(t)}{\beta \Delta t}+\frac{\beta -0.5}{\beta}\mathbf{\ddot{u}}(t) \\ -\mathbf{\dot{u}}(t+ \Delta t) &= \mathbf{\dot{u}}(t)+ (1-\gamma)\Delta t \mathbf{\ddot{u}}(t) + \gamma \Delta t \mathbf{\ddot{u}}(t+\Delta t) -\end{aligned} -\end{equation} + -In the above equations, $\beta$ and $\gamma$ are Newmark time integration parameters. Substituting the above two equations into the equation of motion will result in a linear system of equations ($\mathbf{Au}(t+\Delta t) = \mathbf{b}$) from which $\mathbf{u}(t+\Delta t)$ can be estimated. +!include include/materials/intro_materials-theory.md -For $\beta = 0.25$ and $\gamma = 0.5$, the Newmark time integration scheme is the same as the trapezoidal rule. The trapezoidal rule is an unconditionally stable integration scheme, i.e., the solution does not diverge for any choice of $\Delta t$, and the solution obtained from this scheme is second order accurate. One disadvantage with using trapezoidal rule is the absence of numerical damping to damp out any high frequency numerical noise that is generated due to the discretization of the equation of motion in time. +!include include/materials/linearelasticsoil-theory.md -The Newmark time integration scheme is unconditionally stable for $\gamma \ge \frac{1}{2}$ and $\beta \ge \frac{1}{4}\gamma$. For $\gamma > 0.5$, high frequency oscillations are damped out, but the solution accuracy decreases to first order. +!include include/materials/isoil-theory.md -### Hilber-Hughes-Taylor (HHT) time integration + -The HHT time integration scheme [citep:hughes2000thefinite] is built upon Newmark time integration method to provide an unconditionally stable and second order accurate numerical scheme with the ability to damp out high frequency numerical noise. Here, in addition to the Newmark equations, the equation of motion is also altered resulting in: +!include include/contact/intro_contact-theory.md -\begin{equation} -\rho\mathbf{\ddot{u}}(t+\Delta t) + \nabla \cdot [(1+\alpha) \sigma(t+\Delta t) - \alpha \sigma(t)] = \mathbf{F_{ext}}(t+\alpha\Delta t) -\end{equation} +!include include/contact/thin_layer-theory.md -Here, $\alpha$ is the HHT parameter. The optimum parameter combination to use for this time integration scheme is $\beta = \frac{1}{4}(1-\alpha)^2$, $\gamma = \frac{1}{2} - \alpha$, and $-0.3 \le \alpha \le 0$. + -## Small strain damping +!include include/bcs/intro_bcs-theory.md -When the soil-structure system (including both soil and concrete) -responds to an earthquake excitation, energy is dissipated in two primary -ways: (1)small-strain and hysteretic material damping, and (2) damping due to gapping, -sliding and uplift at the soil-foundation interface. Dissipation of -energy due to item (1) is modeled (approximately) using following methods: (i) viscous damping for small strain damping experienced at very small strain -levels ($\gamma$ $\leq 0.001 \%$) where the material behavior is largely linear viscoelastic; (ii) -hysteretic damping due to nonlinear hysteretic behavior of the material. -Dissipation of energy due to (2) is discussed in [foundation-soil interface models](#Foundation-soil interface models). This section discusses the damping that is present at small strain levels. +!include include/bcs/non_reflecting-theory.md +!include include/bcs/seismic_force-theory.md -### Rayleigh damping +!include include/bcs/preset_acceleration-theory.md -Rayleigh damping is the most common form of classical damping used in modeling structural dynamic problems. The more generalized form of classical damping, Caughey Damping [citep:caughey1960classical], is currently not implemented in MASTODON. Rayleigh damping is a specific form of Caughey damping that uses only the first two terms of the series. In this method, the viscous damping is proportional to the inertial contribution and contribution from the stiffness. This implies that in the matrix form of the governing equation, the damping matrix ($\mathbf{C}$) is assumed to be a linear combination of the mass ($\mathbf{M}$) and stiffness ($\mathbf{K}$) matrices, i.e., $\mathbf{C} = \eta \mathbf{M} +\zeta\mathbf{K}$. Here, $\eta$ and $\zeta$ are the mass and stiffness dependent Rayleigh damping parameters, respectively. + -The equation of motion (in the matrix form) in the presence of Rayleigh damping becomes: -\begin{equation} -\mathbf{M}\mathbf{\ddot{u}}+ (\eta \mathbf{M} + \zeta \mathbf{K})\mathbf{\dot{u}} +\mathbf{K}\mathbf{u} = \mathbf{F_{ext}} -\end{equation} +!include include/fault_rupture/fault_rupture-theory.md -The same equation of motion at any point in space and time (in the non-matrix form) is given by: -\begin{equation} -\rho\mathbf{\ddot{u}} + \eta \rho \mathbf{\dot{u}} + \zeta \nabla \cdot \frac{d}{dt}\sigma + \nabla \cdot \sigma = \mathbf{F_{ext}} -\end{equation} + -The degree of damping in the system depends on the coefficients $\zeta$ and $\eta$ as follows: -\begin{equation}\label{eqn:general_rayleigh} -\xi (f) = \frac{\eta}{2} \frac{1}{f} + \frac{\zeta}{2} f -\end{equation} +!include manuals/include/outputs/intro_outputs-theory.md -where, $\xi(f)$ is the damping ratio of the system as a function of frequency $f$. The damping ratio as a function of frequency for $\zeta = 0.0035$ and $\eta = 0.09$ is presented in [fig:rayleigh]. +!include manuals/include/outputs/responsehistorybuilder-theory.md -!media media/theory/rayleigh.png - style=width:60%;margin-left:150px;float:center; - id=fig:rayleigh - caption=Damping ratio as a function of frequency. +!include manuals/include/outputs/responsespectracalculator-theory.md - -#### Constant damping ratio - -For the constant damping ratio scenario, the aim is to find $\zeta$ and $\eta$ such that the $\xi(f)$ is close to the target damping ratio $\xi_t$, which is a constant value, between the frequency range $[f_1, f_2]$. This can be achieved by minimizing the difference between $\xi_t$ and $\xi(f)$ for all the frequencies between $f_1$ and $f_2$, i.e., if - -\begin{equation} -I = \int_{f_1}^{f_2} \xi_t - \left(\frac{\eta}{2}\frac{1}{f} + \frac{\zeta}{2}f\right) df -\end{equation} - -Then, $\frac{dI}{d \eta} = 0$ and $\frac{dI}{d \zeta}=0$ results in two equations that are linear in $\eta$ and $\zeta$. Solving these two linear equations simultaneously gives: - -\begin{equation} -\begin{aligned} -\zeta &= \frac{\xi_t}{2 \pi} \; \frac{3}{(\Delta f)^2} \; \left(f_1 + f_2 - 2 \frac{f_1 f_1}{\Delta f} \; ln \frac{f_2}{f_1}\right) \\ -\eta &= 2 \pi \xi_t \; \frac{f_1 f_2}{\Delta_f} \; \left[ln \frac{f_2}{f_1}\; \left(2 + 6 \frac{f_1 f_2}{(\Delta_f)^2}\right) - \frac{3(f_1 + f_2)}{\Delta_f}\right] -\end{aligned} -\end{equation} - -#### Damping ratio for soils - -Small strain material damping of soils is independent of loading frequency in frequency band of 0.01 Hz - 10 Hz ([cite:menq2003], [cite:shibuya2000damp],[cite:lopresti1997damp], and [cite:marmureanu2000damp]). The two mode Rayleigh damping is frequency dependent and can only achieve the specified damping at two frequencies while underestimating within and overestimating outside of these frequencies. The parameters $\eta$ and $\zeta$ for a given damping ratio can be calculated as follows: - -\begin{equation} - \begin{bmatrix} - \xi_i \\ - \xi_j - \end{bmatrix} - = - \frac{1}{4\pi} - \begin{bmatrix} - \frac {1}{f_i} & f_i \\ - \frac {1}{f_j} & f_j - \end{bmatrix} - \begin{bmatrix} - \eta \\ - \zeta - \end{bmatrix} -\end{equation} - -In case of two mode Rayleigh damping, [cite:kwok2007damp] suggests to use natural frequency and five times of it for the soil column of interest. In addition, selecting first mode frequency of soil column and higher frequency that corresponds to predominant period of the input ground motion is a common practice. - -Heterogeneities of the wave travel path may introduce scattering effect which leads to frequency dependent damping ([cite:campbell2009damp]). This type of damping is of the form ([cite:withers2015memory]): - -\begin{equation}\label{eqn:non_constant_damping} -\xi (f) = \begin{cases} - \xi_t, & \text{if}\ \; f \le f_T \\ - \xi_t \; \left(\frac{f_T}{f}\right)^\gamma, & \text{if}\ \;f > f_T - \end{cases} -\end{equation} - -where, $f_T$ is the frequency above which the damping ratio starts to -deviate from the constant target value of $\xi_t$, and $\gamma$ is -the exponent which lies between 0 and 1. Minimizing the difference -between [eqn:non_constant_damping] and -[eqn:general_rayleigh] with respect to $\eta$ and $\zeta$ for -all frequencies between $f_1$ and $f_2$ gives: - -\begin{equation} -\begin{aligned} -\zeta &= \frac{\xi_t}{2 \pi} \; \frac{6}{(\Delta f)^3} \; [b(f_1,f_2) - a(f_1, f_2) \; f_1 f_2] \\ -\eta &= 2 \pi \xi_t \; \frac{2 f_1 f_2}{(\Delta f)^3} \; [a(f_1, f_2)\; ({f_1}^2 + {f_2}^2 + f_1 f_2) - 3 b(f_1, f_2)] -\end{aligned} -\end{equation} - -where, the functions $a(f_1, f_2)$ and $b(f_1, f_2)$ are given by: - -\begin{equation} -\begin{aligned} -a(f_1, f_2) &= ln \frac{f_T}{f_1} + \frac{1}{\gamma} \; \left[1- \left(\frac{f_T}{f_2}\right)^\gamma\right] \\ -b(f_1, f_2) &= \frac{{f_T}^2 - {f_1}^2}{2} + \frac{{f_T}^\gamma}{2-\gamma} \; ({f_2}^{2-\gamma} - {f_T}^{2-\gamma}) -\end{aligned} -\end{equation} - -Also, $\xi_t$ for soils is inversely proportional to the shear wave -velocity ($V_s$) and a commonly used expression for $\xi_t$ of soil -is: - -\begin{equation} -\xi_t = \frac{5}{V_s} -\end{equation} - -where, $V_s$ is in m/s. - -### Frequency independent damping - -As seen in the previous sub-section, the damping ratio using Rayleigh damping varies with frequency. Although the parameters $\eta$ and $\zeta$ can be tuned to arrive at a constant damping ratio for a short frequency range, as the frequency range increases, the damping ratio no longer remains constant. For scenarios like these, where a constant damping ratio is required over a large frequency range, frequency independent damping formulations work better. This formulations is under consideration for adding to MASTODON. - -## Soil layers and meshing - -Small strain properties (shear wave velocity, small strain modulus etc.) as well as mobilized shear strength of soils change with depth. Thus, in numerical models, soil profile (layers) is constructed to incorporate the depth dependent properties. The ground surface as well as layers that define the soil domain can be horizontal -or non-horizontal. For the horizontal ground surface and layering scenario, the location -of the interfaces can be provided as input and MASTODON will use that -information to generate a set of soil layers, each with a unique -identification number. These layer ids are later used to assign material -properties to each soil layer. The same procedure can also be used for -non-horizontal but planar soil layers by specifying the normal to the -plane and the interface locations measured along the normal direction. - -For scenarios where the soil layers are non-horizontal and non-planar, images (.jpg, .png, etc.) of the soil profile can be provided as input. The different soil layers are distinguished from the image by reading either the red, green or blue color value (as per user's directions) at each pixel. Gray scale images in which the red, green and blue values are all the same also work well for this purpose. For creating 3D soil layers, multiple 2D images with soil profiles at different 2-D cross-sections of the soil domain can be provided as input. - -Once the soil layers have been distinguished, it is necessary to ensure that the different soil layers are meshed such that they can accurately transmit waves of the required frequency. The optimum element size for each soil layer depends on the type of element used for meshing, cut-off frequency (f) of the wave and the shear wave velocity ($V_s$) of the soil layer. A minimum of 10 points is required per wavelength of the wave to accurately represent the wave in space [citep:coleman2016time]. The minimum wavelength ($\lambda_{min}$) is calculated as: - -\begin{equation} -\lambda_{min} = \frac{V_s}{f} -\end{equation} - -If linear elements such as QUAD4 or HEX8 are used, then the optimum mesh size is $\lambda_{min}/10$. If quadratic elements such as QUAD9 or HEX27 are used, then the optimum mesh size is $\lambda_{min}/5$. Using the minimum element size information, MASTODON refines the mesh such that the element size criterion is met and at the same time the layers separations are visible. An example of this meshing scheme is presented in [fig:adaptive_meshing] where a 2D soil domain is divided into 3 soil layers and these soil layers are meshed such that the element size criterion is satisfied. A denser mesh is created at the interface between different soil layers. - -!media media/theory/adaptive_mesh.png - style=width:60%;margin-left:150px;float:center; - id=fig:adaptive_meshing - caption=Auto-generated mesh for a soil domain with three non-horizontal non-planar soil layers. - -## Material models - -To model the mechanical behavior of a material, three components need to be defined at every point in space and time - strain, elasticity tensor, stress. - -1. **Strain**: Strain is a normalized measure of the deformation experienced by a material. In a 1-D - scenario, say a truss is stretched along its axis, the axial strain is the elongation of the truss - normalized by the length of the truss. In a 3D scenario, the strain is 3x3 tensor and there are - three different ways to calculate strains from displacements - small linearized total strain, - small linearized incremental strain, and finite incremental strain. Details about these methods - can be found in [modules/tensor_mechanics/index.md]. - -2. **Elasticity Tensor**: The elasticity tensor is a 4th order tensor with a maximum of 81 - independent constants. For MASTODON applications, the soil and structure are usually assumed to - behave isotropically, i.e., the material behaves the same in all directions. Under this - assumption, the number of independent elastic constants reduces from 81 to 2. The two independent - constants that are usually provided for the soil are the shear modulus and Poissons's ratio, and - for the structure it is the Young's modulus and Poisson's ratio. - -3. **Stress**: The stress at a point in space and time is a 3x3 tensor which is a function of the - strain at that location. The function that relates the stress tensor to the strain tensor is the - constitutive model. Depending on the constitutive model, the material can behave elastically or - plastically with an increment in strain. - -Details about stress calculation for two different constitutive models are presented below. - -### Linear elastic constitutive model - -In scenarios where the material exhibits a linear relation between stress and strain, and does not -retain any residual strain after unloading, is called a linear elastic material. In linear -elasticity, the stress tensor ($\sigma$) is calculated as $\sigma = \mathcal{C}\epsilon$, where -$\mathcal{C}$ is the elasticity tensor, and $\epsilon$ is the strain tensor. This material model is -currently used for numerically modeling the behavior of concrete and other materials used for -designing a structure in MASTODON. - -### Nonlinear hysteretic constitutive model for soils (I-soil) - -I-soil ([citet:numanoglu2017phd]) is a three dimensional, physically motivated, piecewise linearized -nonlinear hysteretic material model for soils. The model can be represented by shear type -parallel-series distributed nested components (springs and sliders) in one dimensional shear stress -space and its framework is analogous to the distributed element modeling concept developed by -[citet:iwan1967on]. The model behavior is obtained by superimposing the stress-strain response of -nested components. Three dimensional generalization follows [citet:chiang1994anew] and uses von Mises -(independent of effective mean stress) and/or Drucker-Prager (effective mean stress dependent) type -shear yield surfaces depending on user's choice. The yield surfaces are invariant in the stress space -[fig:yield_surface] . Thus, the model does not require kinematic hardening rule to model un/reloading -stress-strain response and preserves mathematical simplicity. - -!media media/theory/yield_surface.png - style=width:40%;margin-left:200px;float:center; - id=fig:yield_surface - caption=Invariant yield surfaces of the individual elastic-perfectly curves - (after Chiang and Beck, 1994). - -The current version of I-soil implemented in MASTODON utilizes Masing type un/reloading behavior and -is analogous to MAT79 (LS-DYNA) material model but does not exhibit numerical instability observed in -MAT79 ([citet:numanoglu2017conf]). Masing type un/reloading is inherently achieved by the model -because upon un/reloading the yielded nested components regain stiffness and strength. The cyclic -response obtained from current version of the model is presented in -[fig:1D_isoil_representation]. Reduction factor type modification on un/reloading behavior -([citet:phillips2009damping]; [citet:numanoglu2017nonmasing]) is an ongoing study within MASTODON -framework. - -!media media/theory/1D_isoil_representation.png - style=width:60%;margin-left:200px;float:center; - id=fig:1D_isoil_representation - caption=I-soil model details: (a) 1D representation by springs; (b) example monotonic and - cyclic behavior of four nested component model (reprinted from Baltaji et al., 2017). - -Main input for the current version of I-soil in MASTODON is a backbone curve at a given reference pressure. MASTODON provides variety of options to built backbone curve for a given soil type and reference pressure using following methods : - -1. User-defined backbone curve (soil\_type = 'user_defined'): The backbone curve can be provided in a - .csv file where the first column is shear strain points and the second column is shear stress - points. The number of nested components that will be generated from this backbone curve depends on - the number of discrete shear strain - shear stress pairs defined in the .csv file. When layered - soil profile is present, .csv file for each reference pressure can be provided to the - corresponding elements in the mesh. - -2. Darendeli backbone curve (soil\_type = 'darandeli'): The backbone curve can be auto-generated - based on empirical relations obtained from laboratory tests. [citet:darendeli2001development] - presents a functional form for normalized modulus reduction curves obtained from resonant column - - torsional shear test for variety of soils. MASTODON utilizes this study and auto-generates the - backbone shear stress - strain curves. The inputs for this option are (1) small strain shear - modulus, (2) bulk modulus, (3) plasticity index, (4) overconsolidation ratio, (5) reference - effective mean stress ($p_{ref}$) at which the backbone is constructed, and (6) number of shear - stress - shear strain points preferred by user to construct piecewise linear backbone curve. All - the other parameters except the number of shear stress - shear strain points can be provided as a - vector for each soil layer. - - Darendeli (2001) study extrapolates the normalized modulus reduction curves after 0.1% shear - strains. This extrapolation causes significant over/under estimation of the shear strength implied - at large strains for different type of soils at different reference effective mean stresses - [citep:hashash2010recent]. Thus user should be cautious about implied shear strength when - utilizing this option. - -3. General Quadratic/Hyperbolic (GQ/H) backbone curve (soil\_type = 'gqh'): - [citet:darendeli2001development] study presented in previous item constructs the shear stress - - shear strain curves based on experimentally obtained data. At small strains, the data is obtained - using resonant column test, and towards the moderate shear strain levels, torsional shear test - results are used. Large strain data are extrapolation of the small to medium shear strain - data. This extrapolation underestimates or overestimates the shear strength mobilized at large - shear strains. Therefore, implied shear strength correction is necessary to account for the - correct shear strength at large strains. GQ/H constitutive model proposed by - [citet:groholski2016simplified] has a unique curve fitting scheme embedded into the constitutive - model that accounts for mobilized strength at large shear strains by controlling the shear - strength. This model uses taumax, theta\_1 through 5, small strain shear modulus, bulk modulus and - number of shear stress - shear strain points preferred by user to construct piecewise linear - backbone curve. The parameter taumax is the maximum shear strength that can be mobilized by the - soil at large strains. The parameters theta\_1 through 5 are the curve fitting parameters and can - be obtained using DEEPSOIL [citep:hashash2016deepsoil]. Other than the number of points, all the - other parameters can be given as a vector for the different soil layers. The number of points, - which determines the number of elastic-perfectly plastic curves to be generated, is constant for - all soil layers. - -Once the backbone curve is provided to I-soil, the model determines the properties for nested -components presented in [fig:1D_isoil_representation]. The stress integration for each nested -component follows classic elastic predictor - plastic corrector type radial return algorithm -([citet:simo2006computein]) and model stress is obtained by summing the stresses from each nested -component: - -\begin{equation} -\tau = \sum_{k=1}^{i} G_{k}*\gamma + \sum_{k=i+1}^{n} {\tau_{y}}^k -\end{equation} - -where $\tau$ is the total shear stress, $G_{k}$ is the shear modulus of the $k^{th}$ nested -component, $\gamma$ is the shear strain, ${\tau_{y}}^k$ is the yield stress of the $k^{th}$ nested -component, and $i$ represents the number of components that have not yet yielded out of the $n$ total -nested components. - -Small strain shear modulus can be varied with effective mean stress via: - -\begin{equation} - G(p) = G_0 \left(\frac{p-p_0}{p_{ref}}\right)^{b_{exp}} -\end{equation} - - -where, $G_0$ is the initial shear modulus, $p_0$ is the tension pressure cut off and $b_{exp}$ is -mean effective stress dependency parameter obtained from experiments. The shear modulus reduces to -zero for any mean effective stress lower than $p_0$ to model the failure of soil in tension. Note -that the mean effective stress is positive for compressive loading. Thus, $p_0$ should be inputted as -negative. - -Yield criteria of the material can also be varied with effective mean stress dependent behavior as: - -\begin{equation} -\tau_y(p) = \sqrt{\frac{a_0 + a_1 (p-p_0) + a_2 (p-p_0)^2}{a_0 + a_1 p_{ref} + a_2 {p_{ref}}^2}} \tau_y(p_{ref}) -\end{equation} - -where, $a_0$, $a_1$ and $a_2$ are parameters that define how the yield stress varies with pressure. - -## Foundation-soil interface models - -The foundation-soil interface is an important aspect of NLSSI modeling. The foundation-soil interface -simulates geometric nonlinearities in the soil-structure system: gapping (opening and closing of gaps -between the soil and the foundation), sliding, and uplift. - -### Thin-layer method - -An efficient approach to modeling the foundation-soil interface is to create a thin layer of the -I-Soil material at the interface, as illustrated in [fig:thin_layer] below. - -!media media/theory/thin_layer.png - style=width:60%;margin-left:100px;float:center; - id=fig:thin_layer - caption=Modeling the foundation-soil interface as a thin layer for a sample surface foundation. - -The red layer between the foundation (green) and soil (yellow) is the thin layer of I-Soil. The -properties of this thin layer are then adjusted to simulate Coulomb friction between the -surfaces. The Coulomb-friction-type behavior can be achieved by modeling the material of the thin -soil layer as follows: - -1. Define an I-Soil material with a user-defined stress-strain curve. - -2. Calculate the shear strength of the thin layer as $\tau_{max}=\mu \sigma_N$ , where $\tau_{max}$ - is the shear strength, $\mu$ is the friction coefficient, and $\sigma_N$ is the normal stress on - the contact surface. The shear strength of the thin layer is the stress at which sliding starts at - the interface. Therefore, this shear strength should be proportional to the normal stress to - simulate Coulomb friction. This can be achieved by setting the initial shear strength equal to the - reference pressure, $p_{ref}$. The reference pressure can then be set to an arbitrary positive - value, such as the average normal stress at the interface from gravity loads. The shear strength - will then follow the equation - - \begin{equation} - \tau_{max} = \mu p_{ref} - \end{equation} - -3. Define the stress-strain curve to be almost elastic-perfectly-plastic, and such that the shear - modulus of the thin layer is equal to the shear modulus of the surrounding soil, in case of an - embedded foundation. If the foundation is resting on the surface such as in [fig:thin_layer] - above, the shear modulus of the thin layer soil should be as high as possible, such that the - linear horizontal foundation stiffness is not reduced due to the presence of the thin layer. A - sample stress-strain curve is shown in [fig:thin_layer_stress_strain] below. The sample curve in - the figure shows an almost bilinear shear behavior with gradual yielding and strain hardening, - both of which, are provided to reduce possible high-frequency response. High-frequency response is - likely to occur if a pure Coulomb friction model (elastic-perfectly-plastic shear behavior at the - interface) is employed, due to the sudden change in the interface shear stiffness to zero. - -!media media/theory/thin_layer_stress_strain.png - style=width:60%;margin-left:150px;float:center; - id=fig:thin_layer_stress_strain - caption=Sample shear-stress shear-strain curve for modeling the thin-layer interface using I-Soil. - -4. Turn on pressure dependency of the soil stress-strain curve and set $a_0$, $a_1$ and $a_2$ to 0, 0 - and 1, respectively. This ensures that the stress-strain curve scales linearly with the normal - pressure on the interface, thereby also increasing the shear strength in the above equation - linearly with the normal pressure, similar to Coulomb friction. - -5. Use a large value for the Poisson’s ratio, in order to avoid sudden changes in the volume of the - thin-layer elements after the yield point is reached. A suitable value for the Poisson’s ratio can - be calculated by trial and error. - -Following the above steps should enable the user to reasonably simulate geometric -nonlinearities. These steps will be automated in MASTODON in the near future. - -## Special boundary conditions - -### Non-reflecting boundary - -This boundary condition applies a Lsymer damper [citep:lysmer1969finite] on a given boundary to -absorb the waves hitting the boundary. To understand Lsymer dampers, let us consider an uniform -linear elastic soil column and say a 1D vertically propagating P wave is traveling through this soil -column. Then the normal stress at any location in the soil column is given by: - -\begin{equation} -\label{eqn:normal_stress} -\sigma = E \epsilon = E \frac{du}{dx} = \frac{E}{V_p} \frac{du}{dt}= \rho V_p \frac{du}{dt} -\end{equation} - -where, $E$ is the Young's modulus, $\sigma$ is the normal stress, $\epsilon$ is the normal strain, -$\rho$ is the density, $V_p = \sqrt{\frac{E}{\rho}}$ is the P-wave speed and $\frac{du}{dt}$ is the -particle velocity. Note that for a 3D problem, the P-wave speed is $V_p = \sqrt{\frac{E(1-\ -nu)}{(1+\nu)(1-2\nu)}}$. - -The stress in the above equation is directly proportional to the particle velocity which makes this -boundary condition analogous to a viscous damper with damping coefficient of $\rho V_p$. So -truncating the soil domain and placing this damper at the end of the domain is equivalent to -simulating wave propagation in an infinite soil column provided the soil is made of linear elastic -material and the wave is vertically incident on the boundary. - -If the soil is not linear elastic or if the wave is incident at an angle on the boundary, the waves -are not completely absorbed by the Lsymer damper. However, if the non-reflecting boundary is placed -sufficiently far from the region of interest, any reflected waves will get damped out by rayliegh -damping or hysteretic material behavior before it reaches the region of interest. - -### Seismic force - -In some cases, the ground excitation is measured at a rock outcrop (where rock is found at surface -level and there is no soil above it). To apply this to a location where rock is say $10$m deep and -there is soil above it, a sideset is created at $10$m depth and the ground excitation (converted into -a stress) is applied at this depth. To apply ground excitation as a stress, the input function should -be given as ground velocity. - -To convert a velocity applied normal to the boundary into a normal stress, Equation -[eqn:normal_stress] can be used. Using a similar argument as discussed in the section above, to -convert a velocity applied tangential to the boundary into a shear stress, Equation -[eqn:shear_stress] can be used. - -\begin{equation} -\label{eqn:shear_stress} -\tau = \rho V_s \frac{du}{dt} -\end{equation} -where, $V_s$ is the shear wave speed and $\tau$ is the shear stress. - -In some situations, the ground motion measured at a depth within the soil is available. This ground -motion is the summation of the wave that enters and exits the soil domain. MASTODON has the -capability to extract the incoming wave from the within soil ground motion. To calculate the incoming -wave velocity, an iterative procedure is used. The initial guess for the incoming wave velocity -($v_i$) at time t is taken to be the same as the within soil velocity measured at that location. The -velocity at this boundary obtained from MASTODON ($v_{mastodon}$) is now going to be different from -the measured within soil velocity ($v_{measured}$) at time t. Half the difference between -$v_{mastodon}$ and $v_{measured}$ is added to $v_o$ and the iterations are continued until $v_i$ -converges (within a numerical tolerance). - -### Preset acceleration - -If the ground excitation was measured at a depth within the soil by placing an accelerometer at that -location, then it is termed as a within-soil input. This time history contains the wave that was -generated by the earthquake (incoming wave) and the wave that is reflected off the free surface. This -ground excitation time history is usually available in the form of a acceleration time history. Since -MASTODON is a displacement controlled algorithm, i.e., displacements are the primary unknown -variables, the acceleration time history is first converted to the corresponding displacement time -history using Newmark time integration equation (Equation [eqn:Newmark]). This displacement time -history is then prescribed to the boundary. - -### Domain reduction method (DRM) - -Earthquake 'source-to-site' simulations require simulating a huge soil domain (order of many -kilometers) with a earthquake fault. The nuclear power plant structure, which is usually less than -100 m wide, is located very far from the earthquake fault, and the presence of the structure only -affects the response of the soil in the vicinity of the structure. In most of these situations, where -a localized feature such as a structure is present in a huge soil domain, the problem can be divided -into two parts: (i) a free-field 'source-to-site' simulation is run on the huge soil domain ( -[fig:DRM](a)) that does not contain the localized feature, and (ii) the forces from the free-field -simulation at one element layer, which is the element layer separating the bigger and smaller soil -domain, can be transferred to a much smaller domain containing the localized feature ( -[fig:DRM](b)). This method of reducing the domain is called the domain reduction method (DRM) -[citep:bielak2003domain]. [fig:DRM] is reproduced from [citet:bielak2003domain]. - -!media media/theory/DRM.png - style=width:100%;float:center; - id=fig:DRM - caption=Domain reduction method summary: (a) Big soil domain containing the earthquake fault - but not the localized feature. The displacements are obtained at the boundaries - $\Gamma$ and $\Gamma_e$ and are converted to equivalent forces. (b) Smaller soil - domain containing the localized feature but not the earthquake fault. The equivalent - forced calculated in (a) are applied at the boundaries $\Gamma$ and $\Gamma_e$. - -To convert the displacements at $\Gamma$ and $\Gamma_e$ from part (i) to equivalent forces, a finite -element model of the one element layer between $\Gamma$ and $\Gamma_e$ is simulated in two -steps. First, the boundary $\Gamma_e$ is fixed and the boundary $\Gamma$ is moved with the -displacements recorded at $\Gamma$. This step gives the equivalent forces at $\Gamma_e$. Second, the -boundary $\Gamma$ is fixed and the boundary $\Gamma_e$ is moved with the displacements recorded at -$\Gamma_e$. This steps gives the equivalent forces at $\Gamma$. - -Note: The meshes for the bigger soil domain and smaller soil domain need not align between $\Gamma$ -and $\Gamma_e$. The equivalent forces can be applied as point forces at the same coordinate location -at which nodes are present in the bigger model, irrespective of whether these locations correspond to -nodal locations in the smaller model. - -## Earthquake fault rupture - -The orientation of an earthquake fault is described using three directions - strike ($\phi_s$), dip -($\delta$) and slip direction ($\lambda$) as shown in [fig:fault_orientation], which is courtesy of -[citet:aki2012quantitative]. - -!media media/fault_orientation.png - style=width:80%;margin-left:100px; - id=fig:fault_orientation - caption=Definition of the fault-orientation parameters - strike $\phi_s$, dip $\delta$ and - slip direction $\lambda$. The slip direction is measured clockwise around from north, - with the fault dipping down to the right of the strike direction. Strike direction is - also measured from the north. $\delta$ is measured down from the horizontal. - -In MASTODON, earthquake fault is modeled using a set of point sources. The seismic moment ($M_o$) of -the earthquake point source in the fault specific coordinate system is: - -\begin{equation} -M_o(t) = \mu A \bar{u}(t) -\end{equation} - -where, $\mu$ is the shear modulus of the soil, $A$ is the area of fault rupture and $\bar{u}(t)$ is -the spatially averaged slip rate of the fault. - -When this seismic moment is converted into the global coordinate system (x, y and z) with the x -direction oriented along the geographic north and z direction along the soil depth, the resulting -moment can be written in a symmetric $3 \times 3$ matrix form whose components are as follows: - -\begin{equation} -\begin{aligned} -M_{xx}(t) &= -M_o(t)(\sin \delta \cos \lambda \sin2 \phi_s + \sin 2\delta \sin\lambda \sin^2 \phi_s) \\ -M_{xy}(t) &= M_{yx}(t) = M_o(t)(\sin\delta \cos \lambda \cos 2 \phi_s + \frac{1}{2} \sin 2\delta \sin \lambda \sin 2 \phi_s) \\ -M_{xz}(t) &= M_{zx}(t) = -M_o(t)(\cos \delta \cos \lambda \cos \phi_s + \cos 2\delta \sin \lambda \sin 2\phi_s) \\ -M_{yy}(t) &= M_o(t)(\sin \delta \cos \lambda \sin 2 \phi_s - \sin 2 \delta \sin \lambda \cos^2 \phi_s) \\ -M_{yz}(t) &= M_{zy}(t) = -M_o(t)(\cos \delta \cos \lambda \sin \phi_s - \cos 2\delta \sin\lambda \cos\phi_s) \\ -M_{zz}(t) &= M_o(t) \sin 2\delta \sin \lambda -\end{aligned} -\end{equation} - -Each component of the above matrix is a force couple with the first index representing the force -direction and the second index representing the direction in which the forces are separated (see -[fig:source_direction]; [citet:aki2012quantitative]). - -!media media/source_direction.png - style=width:60%;margin-left:150px; - id=fig:source_direction - caption=The nine different force couples required to model an earthquake source. - -The total force in global coordinate direction $i$ resulting from an earthquake source applied at -point $\vec{\zeta}$ in space is then: - -\begin{equation} -f_i(\vec{x}, t) = - \sum_{j=1}^{3} \frac{\partial M_{ij}(\vec{x}, t)}{\partial x_j} = \sum_{j=1}^{3} M_{ij}(t) \frac{\partial \delta (\vec{x} - \vec{\zeta})}{\partial x_j} -\end{equation} -where, $\delta(.)$ is the delta function in space. - -When many earthquake sources are placed on the earthquake fault, and they rupture at the same time -instant, then an approximation to a plane wave is generated. If one of the point sources is specified -as the epicenter and the rupture speed ($V_r$) is provided, then the other point sources start -rupturing at $d/V_r$, where $d$ is the distance between the epicenter and the other point source. - -## Post-processing - -This section presents some of the common post-processing tools available in MASTODON that help in -understanding the wave propagation, and response of structures and soils to earthquake excitation. - -### Time histories - -In MASTODON, the time history of any nodal variable (displacement, velocity, acceleration) or -elemental variable (stress, strain) can be requested as output. The nodal variable time histories can -be requested at a set of nodes and this can help in visualizing the wave propagation. - -### Response spectra - -An important quantity that is used in understanding the response of a structure is the -velocity/acceleration response spectra. This contains information about the frequency content of the -velocity/acceleration at a particular location. The velocity/acceleration response spectra at a -frequency $\omega$ is obtained by exciting a single degree of freedom (SDOF) system with natural -frequency of $\omega$ with the velocity/acceleration time history recorded at that location, and -obtaining the peak velocity/acceleration experienced by the SDOF. This exercise is repeated for -multiple frequencies to obtain the full response spectra. - -### Housner spectrum intensity - -The response spectra is very useful for understanding the response of the system at one -location. However, if the response at multiple locations have to be compared, a single value that can -summarize the response at a location is much more useful. Housner spectrum intensity is the integral -of the velocity response spectra between 0.25-2.5 s (or 0.4-4 Hz). This packs the information from -the velocity response spectra at multiple frequencies into a single value and reasonably represents -the response at a location. +!include manuals/include/outputs/hsi-theory.md !bibtex bibliography diff --git a/doc/content/manuals/user/index.md b/doc/content/manuals/user/index.md index 29f222fc7c..8b95d99fe1 100644 --- a/doc/content/manuals/user/index.md +++ b/doc/content/manuals/user/index.md @@ -1,272 +1,33 @@ # User Manual -## Getting Started + -This section provides step-by-step instructions to define the basic components of a simple numerical -model for a finite element analysis using MASTODON framework focusing on the geotechnical earthquake -engineering applications. +!include include/misc/getting_started-user.md -### Defining a simple domain + -The first thing necessary to run a finite element analysis is a meshed domain. Meshing the domain of -interest for MASTODON finite element analysis can be achieved using the inbuilt mesh generator or -using a preprocessor Cubit . For any analysis, the number -of dimensions and the degrees of freedom should be specified in the beginning of the input file. An -example commands are provided below: +!include include/mesh/meshing-user.md -!listing test/tests/materials/isoil/HYS_darendeli.i - start=Mesh - end=GlobalParams + -This command generates a single brick element using the MASTODON mesh generator with 8 nodes, 8 gauss -quadrature points, and unit length in each edge. dim = 3 command specifies that the domain in three -dimensions, and nx, ny, and nz specify the number of elements in the corresponding directions. +!include include/bcs/bcs-user.md -[fig:single_element] presents a generic three-dimensional brick element along with node and surface -labels. Generated brick element is automatically assigned a block number (block 0 in this case) and -each side of the brick is automatically assigned a surface number. + -!media media/user/single_element.png - style=width:60%;margin-left:150px;float:center; - id=fig:single_element - caption=Single brick element (a) node labels (b) surface labels. +!include include/gravity/gravity-user.md -The next step is activating the global parameters. These are input parameters, which have the same -value across the input file. The global parameters are activated providing the following commands: + -!listing test/tests/materials/isoil/HYS_darendeli.i - start=GlobalParams - end=Variables +!include include/executioners/executioner-user.md -By providing the above commands, the user specifies that if any object in the input file has a -parameter called “displacements”, that parameter would be set to “disp_x disp_y disp_z”, which are -the global displacement degrees of freedoms at the nodal points in x, y, and z directions, -respectively. + -Before or after (the sequence does not matter in the input file) the global parameters, the user has -to specify the solution variables using the following commands: +!include include/materials/intro_materials-user.md -!listing test/tests/materials/isoil/HYS_darendeli.i - start=Variables end=AuxVariables +!include include/materials/linearelasticsoil-user.md -MASTODON allows user to define auxiliary variables. Auxiliary variables are not solved for by the -system but they are calculated from the solution variables. This is particularly necessary for -dynamic analysis, since the nodal accelerations and velocities are defined in this section. In -addition, the stresses and strains are defined as auxiliary variables to access the data later on to -inspect the results. Following commands are the examples of defining the auxiliary variables: +!include include/materials/isoil-user.md -!listing test/tests/materials/isoil/HYS_darendeli.i - start=AuxVariables - end=Kernels + -Using the above commands, user defines velocity, acceleration, stress, and strain auxiliary variables -to be used later on to ask for an output. order = CONSTANT and family=MONOMIAL calculates the average -stress within the element. - -The above command blocks forms the basis for all quasi-static and dynamic analyses that can be -conducted using MASTODON framework. Next section describes the Kernels related to physics involved in -the particular analysis. - -### Kernels - -Kernels are related to the physics involved in the particular analysis. This document focusses on -kernel commands related to the geotechnical earthquake engineering applications. Further information -can be found in [/Kernels/index.md]. - -The main kernel that is used for quasi-static and dynamic analyses is -[modules/tensor_mechanics/index.md]. This kernel is used to solve the equation of motion without the -inertial effects. It requires information about the unknowns that are solved for. The following chunk -of commands can be used to activate dynamic tensor mechanics kernel along with Newmark-beta -integration scheme, inertial effects, and two mode Rayleigh viscous damping (both stiffness and mass -proportional damping): - -!listing test/tests/materials/isoil/HYS_darendeli.i - start=Kernels - end=BCs - -displacements = 'disp_x disp_y disp_z' line is not necessary if the displacements are already defined -as global parameters but are provided here for the sake of completeness. Beta and gamma are the -parameters of Newmark-beta integration scheme. Zeta and eta are the stiffness and mass matrix -coefficients of Rayleigh damping formulation (see [modules/tensor_mechanics/index.md] and -[Theory Manual](manuals/theory/index.md) for more information). Auxiliary kernels are specified to -calculate the acceleration and velocities using Newmark-beta scheme at the end of each time step -where the displacement is already solved and known. Lastly, stress_xy is defined as an auxiliary -variable. This is achieved by specifying the type of the Auxkernel as RankTwoAux. RankTwoAux means -that the source of the auxiliary variable is a rank two tensor, and the type of rank two tensor is -explicitly defined as stress tensor using the command “rank_two_tensor = stress”. Since, the variable -is stress_xy, the location in the stress tensor corresponding to stress_xy needs to be specified -using index_i (row index) and index_j (column index). To request for stress_xy, index_i is set to 0 -and index_j is set to 1. The next section explains the boundary conditions that are required to run a -simple, quasi-static and dynamic analyses. - -### Boundary Conditions - -Boundary conditions are required by MASTODON finite element analysis framework to be able to run -simple quasi-static and dynamic analysis. Herein, the basic boundary conditions are given and other -constraints that are used to solve for geotechnical earthquake engineering problems are presented -separately (e.g. periodic boundary conditions.) Following commands are related to the single element -presented in [fig:single_element] and a chunk of command is provided below to demonstrate how to fix -the bottom nodes in all directions. - -!listing test/tests/materials/isoil/HYS_darendeli.i - start=BCs - end=Periodic - -In the above code, type = PresetBC sets the variable = disp_”related degree of freedom” with the -value = 0 which provides the fixity by defining a zero displacement at the node. boundary = 0 command -selects the nodes at the bottom surface of the element (labeled as surface 0) and assigns the -boundary conditions. - -### Analysis Executioner - -In order to unfold the analysis, executioner commands are required. There are two main types of -executioner in MASTODON, which are steady and transient. Herein, an example is provided to run -transient analysis that will be used later on examples related to geotechnical earthquake engineering -applications. - -!listing test/tests/materials/isoil/HYS_darendeli.i - start=Executioner - end=Postprocessors - -The type = Transient command is self-explanatory and sets the executioner type to -transient. solve_type = PJFNK is the default option (preconditioned Jacobian Free Newton-Krylov) used -to solve the transient problems in MASTODON without constructing full stiffness matrix. The command -dt = 0.01 (is set to 0.01 sec. here for an example purpose) determines the initial time step and user -is allowed to set a minimum time step “dtmin” for the analysis. start_time and end_time specifies -where the transient analysis will start and where it will end. Further information on available -options can be found in [/Executioner/index.md]. - -### Outputs - -The analysis results can be dumped to a csv file using postprocessor and output commands. The first -command should define the type of results that will be dumped into the csv file. The following -command is an example to specify the results to be outputted: - -!listing test/tests/materials/isoil/HYS_darendeli.i - start=Postprocessors - -Since, the stresses, accelerations, and velocities are defined as an auxiliary variable previously -([Kernels](manuals/user/index.md)), they can be directly called in this section. elementid = 0 -specifies the element number from which stress values will be pulled out. Note that MASTODON assigns -numbers to elements and nodes different than those of the input file provided by the user (for -user-defined mesh). To obtain the correct element and nodes numbers, the software Blot installed with -MOOSE or Paraview should be used. type = ElementVariableValue returns the average scalar stress value -computed within the specified block. Finally, the results can be dumped into a csv file by using -"Outputs command". Once the analysis is completed, the output csv file will be stored in the same -folder as the input file. The first column contains the “Time”, and the rest of the columns contain -the values asked in the "Postprocessors" command. Another form of output is the exodus file which can -be read by Blot. The user also has the option to request results in both csv and exodus formats. - -### Gravity Loading - -Gravity loading can be applied on the domain to achieve the stress state due to gravity. MOOSE -includes built-in commands to apply the gravity as a body force throughout the transient -analysis. The following command can be used to apply the gravitational field on the modeled domain: - -!listing test/tests/materials/isoil/HYS_darendeli.i - start=gravity - end=AuxKernels - -The above command should be embedded inside the [Kernels](manuals/user/index.md) command -block. variable = disp_z activates the gravity in z direction and value = -9.81 is assigned to -specify the magnitude and direction of the gravitational field. - -Gravity command applies an external body force along the specified direction on the domain of -interest. In order to bring the system to equilibrium, a transient analysis with some viscous damping -is necessary. This approach causes fluctuations on the stress and strains at the beginning of the -analysis because of the elements being initially at zero stress state. Viscous damping removes the -fluctuations and brings the system to equilibrium. Once the system equilibrates, the gravity stresses -are obtained along with the displacements due to the gravity loading. “Initial Stress” command is -available in MASTODON framework to eliminate the need for a separate transient analysis. In addition, -no displacements, or strains due to gravity result. - -### Loads - -There are two types of loads that can be applied to the domain of interest. These are prescribed -displacement and prescribed loads. Both types can be defined as a time history. - -#### Prescribed Displacement - -The preset displacement boundary condition can be used to apply a displacement time history to a -boundary (at the nodes). The displacement boundary condition first converts the user defined -displacement time history to an acceleration time history using Backward Euler finite difference -scheme. This acceleration is then integrated to get displacement using Newmark-beta method.  The -resulting displacement is then applied as a kinematic displacement boundary condition. The following -command can be used to apply the preset displacement boundary condition: - -!listing test/tests/materials/isoil/HYS_darendeli.i - start=top_x - end=Functions - -The above command should be embedded inside the BCs command block. “boundary = 5” assigns the preset -displacement to boundary 5 which, in this case, is a predefined boundary of a single element as -described in [fig:single_element]. Alternatively, the boundary number can be identified using Trelis . “variable = disp_x” imposes the boundary condition on the x -direction. “beta” is the Newmark-beta integration parameter. The “function = top_disp” specifies the -function that defines the loading time history. It is defined in the “Functions” block as follows: - -!listing test/tests/materials/isoil/HYS_darendeli.i - start=Functions - end=Materials - -Displacement2.csv is the file, located in the same directory of the input file, containing the -displacement time history. The first column of this file should contain the time vector starting at -0.0. The second column should contain the displacement values. “type = PiecewiseLinear” defines the -type of the function which is in this case piecewise-linear. “format” specifies the format of the -data file, i.e. whether the data is in columns or rows. - -#### Prescribed Acceleration - -The preset acceleration boundary condition can be used to apply an acceleration time history to a -boundary. The preset acceleration boundary condition integrates the given acceleration time history -to get the displacement using Newmark-beta method. This displacement is then applied as a kinematic -displacement boundary condition. Syntax is the same as prescribing a displacement boundary condition -but with type = PresetAcceleration and the function describing time vs acceleration data instead of -time vs displacement. - -### Periodic Boundary Conditions - -Periodic boundary conditions are used to constrain the nodes to move together in the specified -directions. The following command is an example applied on the element presented in -[fig:single_element] and should be embedded into the BCs block segment as: - -!listing test/tests/materials/isoil/HYS_darendeli.i - start=Periodic - end=top_x - -### Initial Stresses - -Initial stress command is available to apply the prescribed initial stresses to the finite element -domain. If the user applies initial stress state that is equal to that of gravity loadings, upon -transient analysis, no vibration occurs and the domain reaches equilibrium with zero strains. This -eliminates the necessity of conducting separate analysis for free-field problems and allows the user -to start directly with a dynamic analysis such as base shaking etc. The following command can be used -within the material definition to activate the initial stresses in the material block: - -``` -initial_stress = '-4.204286 0 0 0 -4.204286 0 0 0 -9.810' -``` - -The nine numerical values fill the stress tensor in the following order - xx, xy, xz, yz, yy, yz, zx, -zy, and zz. The above example command activates -4.2025 units of stress along both xx and yy -direction, and -9.81 units of stress along zz direction. These stresses are the gravity stresses at -the mid-depth of the element in [fig:single_element] with density = 2. Therefore, the element will -immediately start with gravity stress conditions without any elastic deformations or stress -fluctuations. - -### Materials - -#### Linear elastic soil - -A linear elastic soil can be defined in MASTODON by defining a elasticity tensor using -ComputeIsotropicElasticityTensor, stress calculator using ComputeLinearElasticStress, and a small -strain calculator using ComputeSmallStrain. Density can be defined using the -GenericConstantMaterial. An example of these input blocks is shown below: - -!listing test/tests/dirackernels/seismic_source/one_seismic_source.i - start=Materials - end=Executioner - -#### I-soil material model - -The nonlinear soil material model usage is demonstrated in the [Example 1](examples/index.md) +!include include/outputs/intro_outputs-user.md diff --git a/doc/content/source/actions/ISoilAction.md b/doc/content/source/actions/ISoilAction.md index 86f60a0dab..5bc9eb6438 100644 --- a/doc/content/source/actions/ISoilAction.md +++ b/doc/content/source/actions/ISoilAction.md @@ -2,4 +2,10 @@ !syntax description /Materials/I_Soil/ISoilAction -Please refer to the [I_Soil](I_Soil/index.md) page. +## Description + +This action is used to set up the I-soil material model for a set of soil layers. + +!syntax parameters /Materials/I_Soil/ISoilAction + +!bibtex bibliography diff --git a/doc/content/source/bcs/NonReflectingBC.md b/doc/content/source/bcs/NonReflectingBC.md index 89e9b1229f..0ce79ca2a0 100644 --- a/doc/content/source/bcs/NonReflectingBC.md +++ b/doc/content/source/bcs/NonReflectingBC.md @@ -4,7 +4,7 @@ ## Description -This boundary condition applies a Lsymer damper [citep:lysmer1969finite] on a given boundary to absorb the waves hitting the boundary. To understand Lsymer dampers, let us consider an uniform linear elastic soil column and say a 1D vertically propagating P wave is traveling through this soil column. Then the normal stress at any location in the soil column is given by: +This boundary condition applies a Lysmer damper [citep:lysmer1969finite] on a given boundary to absorb the waves hitting the boundary. To understand Lysmer dampers, let us consider an uniform linear elastic soil column and say a 1D vertically propagating P wave is traveling through this soil column. Then the normal stress at any location in the soil column is given by: $$ \sigma = E \epsilon = E \frac{du}{dx} = \frac{E}{V_p} \frac{du}{dt}= \rho V_p \frac{du}{dt}, $$ @@ -12,7 +12,7 @@ where $E$ is the Young's modulus, $\sigma$ is the normal stress, $\epsilon$ is t The stress in the above equation is directly proportional to the particle velocity which makes this boundary condition analogous to a viscous damper with damping coefficient of $\rho V_p$. So truncating the soil domain and placing this damper at the end of the domain is equivalent to simulating wave propagation in an infinite soil column provided the soil is made of linear elastic material and the wave is vertically incident on the boundary. -If the soil is not linear elastic or if the wave is incident at an angle on the boundary, the waves are not completely absorbed by the Lsymer damper. However, if the non-reflecting boundary is placed sufficiently far from the region of interest, any reflected waves will get damped out by rayliegh damping or hysteretic material behavior before it reaches the region of interest. +If the soil is not linear elastic or if the wave is incident at an angle on the boundary, the waves are not completely absorbed by the Lysmer damper. However, if the non-reflecting boundary is placed sufficiently far from the region of interest, any reflected waves will get damped out by rayliegh damping or hysteretic material behavior before it reaches the region of interest. !syntax parameters /BCs/NonReflectingBC