[cig-commits] r16883 - in short/3D/PyLith/trunk/doc/userguide: governingeqns runpylith
brad at geodynamics.org
brad at geodynamics.org
Thu Jun 3 16:15:58 PDT 2010
Author: brad
Date: 2010-06-03 16:15:58 -0700 (Thu, 03 Jun 2010)
New Revision: 16883
Removed:
short/3D/PyLith/trunk/doc/userguide/governingeqns/qstatic-smalldef.lyx
Modified:
short/3D/PyLith/trunk/doc/userguide/runpylith/runpylith.lyx
Log:
Added section on different solver types.
Deleted: short/3D/PyLith/trunk/doc/userguide/governingeqns/qstatic-smalldef.lyx
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/governingeqns/qstatic-smalldef.lyx 2010-06-03 22:31:49 UTC (rev 16882)
+++ short/3D/PyLith/trunk/doc/userguide/governingeqns/qstatic-smalldef.lyx 2010-06-03 23:15:58 UTC (rev 16883)
@@ -1,1306 +0,0 @@
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-\begin_document
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-
-\begin_body
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-\begin_layout Section
-Governing Equations
-\end_layout
-
-\begin_layout Standard
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-
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-
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-
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-\begin_inset Text
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-\begin_layout Standard
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-
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-
-\begin_layout Standard
-\begin_inset Formula $u_{i}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\overrightarrow{u}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Displacement vector field
-\end_layout
-
-\end_inset
-</cell>
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-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $f_{i}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\overrightarrow{f}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Body force vector field
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $T_{i}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\overrightarrow{T}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Traction vector field
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\sigma_{ij}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\underline{\sigma}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Stress tensor field
-\end_layout
-
-\end_inset
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-<row bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $n_{i}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\overrightarrow{n}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
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-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Normal vector field
-\end_layout
-
-\end_inset
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-
-\begin_layout Standard
-\begin_inset Formula $\rho$
-\end_inset
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-
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-\begin_inset Formula $\rho$
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-
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-
-
-\end_layout
-
-\begin_layout Subsection
-Constitutive Model
-\end_layout
-
-\begin_layout Subsubsection
-Index Notation
-\end_layout
-
-\begin_layout Standard
-We consider a general class of quasi-static viscoelastic models under the
- assumption of infinitesimal strain, and the methods we derive are also
- appropriate for viscoplastic behavior.
- The stresses are considered to be a function of the total strains and possibly
- of other variables as well:
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \begin{equation}
-\sigma_{ij}=h_{ij}\left(\varepsilon_{ij},q_{k}\right)+\sigma_{ij}^{0}.\end{equation}
-
-\end_inset
-
-The strains are given by
-\begin_inset Formula $\varepsilon_{ij}$
-\end_inset
-
-, while the
-\begin_inset Formula $q_{k}$
-\end_inset
-
- represent additional variables upon which the stress depends.
- These additional variables follow the evolution equations
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \begin{equation}
-\dot{q_{k}}=r_{k}\left(\varepsilon_{ij},q_{k}\right),\end{equation}
-
-\end_inset
-
-with the initial conditions
-\begin_inset Formula $q_{k}\left(t_{0}\right)=q_{k}^{0}$
-\end_inset
-
-.
- The
-\begin_inset Formula $\sigma_{ij}^{0}$
-\end_inset
-
- are the initial stresses in the material.
- To simplify our derivations, we consider the initial stresses to be included
- in the additional variables,
-\begin_inset Formula $q_{k}$
-\end_inset
-
-.
- The strain tensor is given by
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \begin{equation}
-\varepsilon_{ij}=\frac{1}{2}\left(u_{i,j}+u_{j,i}\right).\end{equation}
-
-\end_inset
-
-The stress can be either a linear or nonlinear function of strain and the
- additional variables, and in some cases the additional variables are not
- needed.
- In linear elastic behavior in the absence of initial stresses, for example,
- the stress is linearly dependent on the strain and there are no additional
- variables.
-\end_layout
-
-\begin_layout Subsubsection
-Vector Notation
-\end_layout
-
-\begin_layout Standard
-We consider a general class of quasi-static viscoelastic models under the
- assumption of infinitesimal strain, and the methods we derive are also
- appropriate for viscoplastic behavior.
- The stresses are considered to be a function of the total strains and possibly
- of other variables as well:
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \begin{equation}
-\boldsymbol{\underline{\sigma}}=\boldsymbol{\underline{h}}\left(\boldsymbol{\underline{\varepsilon}},\boldsymbol{q}\right)+\boldsymbol{\sigma}_{0}.\end{equation}
-
-\end_inset
-
-The strains are given by
-\begin_inset Formula $\boldsymbol{\underline{\varepsilon}}$
-\end_inset
-
-, while the
-\begin_inset Formula $\boldsymbol{q}$
-\end_inset
-
- represent additional variables upon which the stress depends.
- These additional variables follow the evolution equations
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \begin{equation}
-\dot{\boldsymbol{q}}=\boldsymbol{r}\left(\boldsymbol{\underline{\varepsilon}},\boldsymbol{q}\right),\end{equation}
-
-\end_inset
-
-with the initial conditions
-\begin_inset Formula $\boldsymbol{q}\left(t_{0}\right)=\boldsymbol{q}_{0}$
-\end_inset
-
-.
- The
-\begin_inset Formula $\underline{\boldsymbol{\sigma}}_{0}$
-\end_inset
-
- are the initial stresses in the material.
- To simplify our derivations, we consider the initial stresses to be included
- in the additional variables,
-\begin_inset Formula $\boldsymbol{q}$
-\end_inset
-
-.
- The strain tensor is given by
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \begin{equation}
-\boldsymbol{\underline{\varepsilon}}=\frac{1}{2}\left(\boldsymbol{\bigtriangledown}\boldsymbol{u}+\boldsymbol{\bigtriangledown}^{T}\boldsymbol{u}\right).\end{equation}
-
-\end_inset
-
-The stress can be either a linear or nonlinear function of strain and the
- additional variables, and in some cases the additional variables are not
- needed.
- In linear elastic behavior in the absence of initial stresses, for example,
- the stress is linearly dependent on the strain and there are no additional
- variables.
-\end_layout
-
-\begin_layout Subsection
-Derivation of Equilibrium Equations
-\end_layout
-
-\begin_layout Subsubsection
-Index Notation
-\end_layout
-
-\begin_layout Standard
-Consider volume
-\begin_inset Formula $V$
-\end_inset
-
- bounded by surface
-\begin_inset Formula $S$
-\end_inset
-
-.
- Applying a Lagrangian description of the conservation of momentum (in the
- absence of accelerations) gives
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \begin{equation}
-\int_{V}f_{i}\, dV+\int_{S}T_{i}\, dS=0.\label{eqn:momentum:index}\end{equation}
-
-\end_inset
-
-The traction vector field is related to the stress tensor through
-\begin_inset Formula \begin{equation}
-T_{i}=\sigma_{ij}n_{j},\end{equation}
-
-\end_inset
-
-where
-\begin_inset Formula $n_{j}$
-\end_inset
-
- is the vector normal to
-\begin_inset Formula $S$
-\end_inset
-
-.
- Substituting into
-\begin_inset LatexCommand \eqref{eqn:momentum:index}
-
-\end_inset
-
- yields
-\begin_inset Formula \begin{equation}
-\int_{V}f_{i}\, dV+\int_{S}\sigma_{ij}n_{j}\, dS=0.\end{equation}
-
-\end_inset
-
-Applying the divergence theorem,
-\begin_inset Formula \begin{equation}
-\int_{V}a_{i,j}\: dV=\int_{S}a_{j}n_{j}\: dS,\end{equation}
-
-\end_inset
-
-to the surface integral results in
-\begin_inset Formula \begin{equation}
-\int_{V}f_{i}\, dV+\int_{V}\sigma_{ij,j}\, dV=0,\end{equation}
-
-\end_inset
-
-which we can rewrite as
-\begin_inset Formula \begin{equation}
-\int_{V}\left(f_{i}+\sigma_{ij,j}\right)\, dV=0.\end{equation}
-
-\end_inset
-
-Because the volume
-\begin_inset Formula $V$
-\end_inset
-
- is arbitrary, the integrand must be zero at every location in the volume,
- so that we end up with
-\begin_inset Formula \begin{gather}
-f_{i}+\sigma_{ij,j}=0\text{ in }V,\\
-\sigma_{ij}n_{j}=T_{i}\text{ on }S_{T}\text{, and}\\
-u_{i}=u_{i}^{o}\text{ on }S_{u.}\end{gather}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Note that since both
-\begin_inset Formula $T_{i}$
-\end_inset
-
- and
-\begin_inset Formula $u_{i}$
-\end_inset
-
- are vector quantities, there can be some spatial overlap of the conceptual
- surfaces
-\begin_inset Formula $S_{T}$
-\end_inset
-
- and
-\begin_inset Formula $S_{u}$
-\end_inset
-
-; however, the same degree of freedom cannot simultaneously have both types
- of boundary conditions.
-\end_layout
-
-\begin_layout Subsubsection
-Vector Notation
-\end_layout
-
-\begin_layout Standard
-Consider volume
-\begin_inset Formula $V$
-\end_inset
-
- bounded by surface
-\begin_inset Formula $S$
-\end_inset
-
-.
- Applying a Lagrangian description of the conservation of momentum (in the
- absence of accelerations) gives
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \begin{equation}
-\int_{V}\overrightarrow{f}\, dV+\int_{S}\overrightarrow{T}\, dS=0.\label{eqn:momentum:vec}\end{equation}
-
-\end_inset
-
-The traction vector field is related to the stress tensor through
-\begin_inset Formula \begin{equation}
-\overrightarrow{T}=\underline{\sigma}\cdot\overrightarrow{n},\end{equation}
-
-\end_inset
-
-where
-\begin_inset Formula $\overrightarrow{n}$
-\end_inset
-
- is the vector normal to
-\begin_inset Formula $S$
-\end_inset
-
-.
- Substituting into
-\begin_inset LatexCommand \eqref{eqn:momentum:vec}
-
-\end_inset
-
- yields
-\begin_inset Formula \begin{equation}
-\int_{V}\overrightarrow{f}\, dV+\int_{S}\underline{\sigma}\cdot\overrightarrow{n}\, dS=0.\end{equation}
-
-\end_inset
-
-Applying the divergence theorem,
-\begin_inset Formula \begin{equation}
-\int_{V}\nabla\cdot\overrightarrow{a}\: dV=\int_{S}\overrightarrow{a}\cdot\overrightarrow{n}\: dS,\end{equation}
-
-\end_inset
-
-to the surface integral results in
-\begin_inset Formula \begin{equation}
-\int_{V}\overrightarrow{f}\, dV+\int_{V}\nabla\cdot\underline{\sigma}\, dV=0,\end{equation}
-
-\end_inset
-
-which we can rewrite as
-\begin_inset Formula \begin{equation}
-\int_{V}\left(\overrightarrow{f}+\nabla\cdot\overrightarrow{\sigma}\right)\, dV=0.\end{equation}
-
-\end_inset
-
-Because the volume
-\begin_inset Formula $V$
-\end_inset
-
- is arbitrary, the integrand must be zero at every location in the volume,
- so that we end up with
-\begin_inset Formula \begin{gather}
-\overrightarrow{f}+\nabla\cdot\overrightarrow{\sigma}=0\text{ in }V,\\
-\underline{\sigma}\cdot\overrightarrow{n}=\overrightarrow{T}\text{ on }S_{T}\text{, and}\\
-\overrightarrow{u}=\overrightarrow{u^{o}}\text{ on }S_{u.}\end{gather}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Note that since both
-\begin_inset Formula $\overrightarrow{T}$
-\end_inset
-
- and
-\begin_inset Formula $\overrightarrow{u}$
-\end_inset
-
- are vector quantities, there can be some spatial overlap of the conceptual
- surfaces
-\begin_inset Formula $S_{T}$
-\end_inset
-
- and
-\begin_inset Formula $S_{u}$
-\end_inset
-
-; however, the same degree of freedom cannot simultaneously have both types
- of boundary conditions.
-\end_layout
-
-\begin_layout Subsection
-Finite-Element Formulation of Quasi-Static Equations
-\end_layout
-
-\begin_layout Subsubsection
-Index Notation
-\end_layout
-
-\begin_layout Standard
-We start with the strong form for the quasi-static problem under the assumption
- of infinitesimal strains,
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \begin{gather}
-\sigma_{ij,j}+f_{i}=0\text{ in }V,\\
-\sigma_{ij}n_{j}=T_{i}\text{ on }S_{T},\\
-u_{i}=u_{i}^{o}\text{ on }S_{u},\\
-\sigma_{ij}=\sigma_{ji}\text{ (symmetric).}\end{gather}
-
-\end_inset
-
-We construct the weak form by multiplying the equation by a trial function
- and setting the integral over the domain to zero.
- The trial function is a piecewise differential vector field,
-\begin_inset Formula $\phi_{i}$
-\end_inset
-
-, where
-\begin_inset Formula $\phi_{i}=0$
-\end_inset
-
- on
-\begin_inset Formula $S_{u}.$
-\end_inset
-
- Hence our weak form is
-\begin_inset Formula \begin{gather}
-\int_{V}\left(\sigma_{ij,j}+f_{i}\right)\phi_{i}\, dV=0\text{, or }\\
-\int_{V}\sigma_{ij,j}\phi_{i}\: dV+\int_{V}f_{i}\phi_{i}\: dV=0.\end{gather}
-
-\end_inset
-
-Consider the divergence theorem applied to the dot product of the stress
- tensor and the trial function,
-\begin_inset Formula $\sigma_{ij}\phi_{i}$
-\end_inset
-
-,
-\begin_inset Formula \begin{equation}
-\int_{V}(\sigma_{ij}\phi_{i})_{,j}\, dV=\int_{S}(\sigma_{ij}\phi_{i})n_{i}\, dS.\end{equation}
-
-\end_inset
-
-Expanding the left hand side yields
-\begin_inset Formula \begin{gather}
-\int_{V}\sigma_{ij,j}\phi_{i}\: dV+\int_{V}\sigma_{ij}\phi_{i,j}\: dV=\int_{S}\sigma_{ij}\phi_{i}n_{i}\: dS,\text{ or}\\
-\int_{V}\sigma_{ij,j}\phi_{i}\: dV=-\int_{V}\sigma_{ij}\phi_{i,j}\, dV+\int_{S}\sigma_{ij}\phi_{i}n_{i}\, dS.\end{gather}
-
-\end_inset
-
-Substituting into the weak form gives
-\begin_inset Formula \begin{equation}
--\int_{V}\sigma_{ij}\phi_{i,j}\, dV+\int_{S}\sigma_{ij}\phi_{i}n_{i}\, dS+\int_{V}f_{i}\phi_{i}\, dV=0.\end{equation}
-
-\end_inset
-
-Now,
-\begin_inset Formula $\sigma_{ij}\phi_{i,j}$
-\end_inset
-
- is a scalar, so it is symmetric,
-\begin_inset Formula \begin{equation}
-\sigma_{ij}\phi_{i,j}=\sigma_{ji}\phi_{j,i},\end{equation}
-
-\end_inset
-
-and we know that
-\begin_inset Formula $\sigma_{ij}$
-\end_inset
-
- is symmetric, so
-\begin_inset Formula \begin{equation}
-\sigma_{ij}\phi_{i,j}=\sigma_{ij}\phi_{j,i},\end{equation}
-
-\end_inset
-
-which means
-\begin_inset Formula \begin{equation}
-\phi_{i,j}=\phi_{j,i},\end{equation}
-
-\end_inset
-
-which we can write as
-\begin_inset Formula \begin{equation}
-\phi_{i,j}=\frac{1}{2}(\phi_{i,j}+\phi_{j,i}).\end{equation}
-
-\end_inset
-
-Substituting into the first term gives
-\begin_inset Formula \begin{equation}
--\int_{V}\frac{1}{2}\sigma_{ij}\left(\phi_{i,j}+\phi_{j,i}\right)\, dV+\int_{S}\sigma_{ij}\phi_{i}n_{i}\, dS+\int_{V}f_{i}\phi_{i}\, dV=0.\end{equation}
-
-\end_inset
-
-Turning our attention to the second term, we separate the integration over
-
-\begin_inset Formula $S$
-\end_inset
-
- into integration over
-\begin_inset Formula $S_{T}$
-\end_inset
-
- and
-\begin_inset Formula $S_{u}$
-\end_inset
-
-,
-\begin_inset Formula \begin{equation}
--\int_{V}\frac{1}{2}\sigma_{ij}(\phi_{i,j}+\phi_{j,i})\, dV+\int_{S_{T}}\sigma_{ij}\phi_{i}n_{i}\, dS+\int_{S_{u}}\sigma_{ij}\phi_{i}n_{i}\, dS+\int_{V}f_{i}\phi_{i}\, dV=0,\end{equation}
-
-\end_inset
-
-and recognize that
-\begin_inset Formula \begin{gather}
-\sigma_{ij}n_{i}=T_{i}\text{ on }S_{T}\text{ and}\\
-\phi_{i}=0\text{ on }S_{u},\end{gather}
-
-\end_inset
-
-so that the equation reduces to
-\begin_inset Formula \begin{equation}
--\int_{V}\frac{1}{2}\sigma_{ij}(\phi_{i,j}+\phi_{j,i})\: dV+\int_{S_{T}}T_{i}\phi_{i}\, dS+\int_{V}f_{i}\phi_{i}\, dV=0.\end{equation}
-
-\end_inset
-
-This is the equation we want to solve.
- Discretizing into finite-elements separates the integral over the domain
- and boundaries into a sum of integrals over elements and element boundaries,
-\begin_inset Formula \begin{equation}
-\sum_{elements}(\int_{V^{e}}\frac{1}{2}\sigma_{ij}(\phi_{i,j}+\phi_{j,i})\, dV-\int_{V^{e}}f_{i}\phi_{i}\, dV-\int_{S_{t}^{e}}T_{i}\phi_{i}\, dS)=0.\end{equation}
-
-\end_inset
-
-Within an element we represent the fields as a linear combination of a set
- of basis functions and the values of the fields at vertices of the element,
-\begin_inset Formula \begin{equation}
-a_{i}=N^{m}a_{i}^{m},\end{equation}
-
-\end_inset
-
-where
-\begin_inset Formula $N^{m}$
-\end_inset
-
- is the
-\begin_inset Formula $m$
-\end_inset
-
-th basis function for an element and
-\begin_inset Formula $a_{i}^{m}$
-\end_inset
-
- is the field at vertex
-\begin_inset Formula $m$
-\end_inset
-
-.
- Rewriting the trial functions and displacement field in terms of the basis
- functions gives
-\begin_inset Formula \begin{gather}
-\phi_{i}=N^{m},\text{ and}\\
-u_{i}=N^{m}u_{i}^{m}.\end{gather}
-
-\end_inset
-
-We force the weak form to hold for each component in the vector space.
- For basis function
-\begin_inset Formula $N^{p}$
-\end_inset
-
- and component
-\begin_inset Formula $i$
-\end_inset
-
-, we have
-\begin_inset Formula \begin{multline}
-\sum_{elements}(\int_{V^{e}}\frac{1}{2}\sigma_{ij}(N_{,j}^{p}+N_{,i}^{p})\, dV-\int_{V^{e}}N_{}^{p}f_{i}\: dV-\int_{S_{T}^{e}}N_{}^{p}T_{i}\, dS)=0.\end{multline}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsubsection
-Vector Notation
-\end_layout
-
-\begin_layout Standard
-We start with the strong form of the quasi-stati problem under the assumption
- of infinitesimal strains,
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \begin{gather}
-\nabla\cdot\underline{\sigma}+\overrightarrow{f}=0\text{ in }V,\\
-\underline{\sigma}\cdot\overrightarrow{n}=\overrightarrow{T}\text{ on }S_{T},\\
-\overrightarrow{u}=\overrightarrow{u}^{o}\text{ on }S_{u},\\
-\underline{\sigma}=\underline{\sigma}^{T}\text{ (symmetric).}\end{gather}
-
-\end_inset
-
-We construct the weak form by multiplying the wave equation by a trial function
- and setting the integral over the domain to zero.
- The trial function is a piecewise differential vector field,
-\begin_inset Formula $\overrightarrow{\phi}$
-\end_inset
-
-, where
-\begin_inset Formula $\overrightarrow{\phi}=0$
-\end_inset
-
- on
-\begin_inset Formula $S_{u}.$
-\end_inset
-
- Hence our weak form is
-\begin_inset Formula \begin{gather}
-\int_{V}\left(\nabla\cdot\underline{\sigma}+\overrightarrow{f}\right)\cdot\overrightarrow{\phi}\, dV=0\text{, or }\\
-\int_{V}(\nabla\cdot\underline{\sigma})\cdot\overrightarrow{\phi}\: dV+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\: dV=0.\end{gather}
-
-\end_inset
-
- Consider the divergence theorem applied to the dot product of the stress
- tensor and the trial function,
-\begin_inset Formula $\underline{\sigma}\cdot\overrightarrow{\phi}$
-\end_inset
-
-,
-\begin_inset Formula \begin{equation}
-\int_{V}\nabla\cdot(\underline{\sigma}\cdot\overrightarrow{\phi})\, dV=\int_{S}(\underline{\sigma}\cdot\overrightarrow{\phi})\cdot\overrightarrow{n}\, dS.\end{equation}
-
-\end_inset
-
-Expanding the left hand side yields
-\begin_inset Formula \begin{equation}
-\int_{V}(\nabla\cdot\underline{\sigma})\cdot\overrightarrow{\phi}\: dV+\int_{V}\underline{\sigma}:\nabla\overrightarrow{\phi}\: dV=\int_{S}(\underline{\sigma}\cdot\overrightarrow{\phi})\cdot\overrightarrow{n}\: dS,\text{ or}\end{equation}
-
-\end_inset
-
-
-\begin_inset Formula \begin{equation}
-\int_{V}{(\nabla\cdot\underline{\sigma})\cdot\overrightarrow{\phi}}_{ij,j}\: dV=-\int_{V}\underline{\sigma}:\nabla\overrightarrow{\phi}\, dV+\int_{S}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS.\end{equation}
-
-\end_inset
-
-Substituting into the weak form gives
-\begin_inset Formula \begin{equation}
--\int_{V}\underline{\sigma}:\nabla\overrightarrow{\phi}\, dV+\int_{S}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS+\int_{V}\overrightarrow{f}\overrightarrow{\phi}\, dV=0.\end{equation}
-
-\end_inset
-
-Now,
-\begin_inset Formula $\underline{\sigma}:\nabla\overrightarrow{\phi}$
-\end_inset
-
- is a scalar, so it is symmetric,
-\begin_inset Formula \begin{equation}
-{\underline{\sigma}:\nabla\overrightarrow{\phi}=(\underline{\sigma}:\nabla\overrightarrow{\phi})}^{T}=\underline{\sigma}^{T}:\overrightarrow{\phi}^{T}\nabla^{T},\end{equation}
-
-\end_inset
-
-and we know that
-\begin_inset Formula $\underline{\sigma}$
-\end_inset
-
- is symmetric, so
-\begin_inset Formula \begin{equation}
-\underline{\sigma}:\nabla\overrightarrow{\phi}=\underline{\sigma}:\overrightarrow{\phi}^{T}\nabla^{T},\end{equation}
-
-\end_inset
-
-which means
-\begin_inset Formula \begin{equation}
-\nabla\overrightarrow{\phi}=\overrightarrow{\phi}^{T}\nabla^{T},\end{equation}
-
-\end_inset
-
-which we can write as
-\begin_inset Formula \begin{equation}
-\nabla\overrightarrow{\phi}=\frac{1}{2}(\nabla+\nabla^{T})\overrightarrow{\phi}.\end{equation}
-
-\end_inset
-
-Substituting into the first term gives
-\begin_inset Formula \begin{equation}
--\int_{V}\frac{1}{2}\underline{\sigma}:(\nabla+\nabla^{T})\overrightarrow{\phi}\, dV+\int_{S}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\, dV=0.\end{equation}
-
-\end_inset
-
-Turning our attention to the second term, we separate the integration over
-
-\begin_inset Formula $S$
-\end_inset
-
- into integration over
-\begin_inset Formula $S_{T}$
-\end_inset
-
- and
-\begin_inset Formula $S_{u}$
-\end_inset
-
-,
-\begin_inset Formula \begin{multline}
--\int_{V}\frac{1}{2}\underline{\sigma}:(\nabla+\nabla^{T})\overrightarrow{\phi}\, dV+\int_{S_{T}}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS+\int_{S_{u}}\underline{\sigma}\cdot\overrightarrow{n}\cdot\overrightarrow{\phi}\, dS+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\, dV=0\\
-\\\end{multline}
-
-\end_inset
-
-and recognize that
-\begin_inset Formula \begin{gather}
-\underline{\sigma}\cdot\overrightarrow{n}=\overrightarrow{T}\text{ on }S_{T}\text{ and}\\
-\overrightarrow{\phi}=0\text{ on }S_{u},\end{gather}
-
-\end_inset
-
-so that the equation reduces to
-\begin_inset Formula \begin{equation}
--\int_{V}\frac{1}{2}\underline{\sigma}:(\nabla+\nabla^{T})\overrightarrow{\phi}\: dV+\int_{S_{T}}\overrightarrow{T}\cdot\overrightarrow{\phi}\, dS+\int_{V}\overrightarrow{f}\cdot\overrightarrow{\phi}\, dV=0.\end{equation}
-
-\end_inset
-
-This is the equation we want to solve.
- Discretizing into finite-elements separates the integral over the domain
- and boundaries into a sum of integrals over elements and element boundaries,
-\begin_inset Formula \begin{equation}
-\sum_{elements}(\int_{V^{e}}\frac{1}{2}\underline{\sigma}:(\nabla+\nabla^{T})\overrightarrow{\phi}\, dV-\int_{V^{e}}\overrightarrow{f}\cdot\overrightarrow{\phi}\, dV-\int_{S_{t}^{e}}\overrightarrow{T}\cdot\overrightarrow{\phi}\, dS)=0.\end{equation}
-
-\end_inset
-
-Within an element we represent the fields as a linear combination of a set
- of basis functions and the values of the fields at vertices of the element,
-\begin_inset Marginal
-status open
-
-\begin_layout Standard
-Is this written correctly?
-\end_layout
-
-\end_inset
-
-
-\begin_inset Formula \begin{equation}
-\overrightarrow{a}=\underline{N}\cdot\overrightarrow{a^{e}},\end{equation}
-
-\end_inset
-
-where
-\begin_inset Formula $\underline{N}$
-\end_inset
-
- are the basis functions for an element and
-\begin_inset Formula $\overrightarrow{a^{e}}$
-\end_inset
-
- is the field at an element's vertices.
- Rewriting the trial functions and displacement field in terms of the basis
- functions gives
-\begin_inset Marginal
-status open
-
-\begin_layout Standard
-Is the trial function expression correct?
-\end_layout
-
-\end_inset
-
-
-\begin_inset Formula \begin{equation}
-\overrightarrow{\phi}=\overrightarrow{N},\text{ and}\end{equation}
-
-\end_inset
-
-
-\begin_inset Formula \begin{equation}
-\overrightarrow{u}=\underline{N}\cdot\overrightarrow{u^{e}}.\end{equation}
-
-\end_inset
-
-Substituting into the integral equation yields
-\begin_inset Formula \begin{multline}
-\sum_{elements}(\int_{V^{e}}\frac{1}{2}\underline{\sigma}:(\nabla+\nabla^{T})\underline{N}\, dV-\int_{V^{e}}\underline{N}\cdot\overrightarrow{f^{e}}^{}\, dV-\int_{S_{T}}\underline{N}\cdot\overrightarrow{T^{e}}_{}\, dS)=0\end{multline}
-
-\end_inset
-
-For a linearly elastic material
-\begin_inset Formula \begin{equation}
-\underline{\sigma}=\underline{C}\cdot\underline{\varepsilon},\end{equation}
-
-\end_inset
-
-and for infinitesimal strains
-\begin_inset Formula \begin{equation}
-\underline{\varepsilon}=\frac{1}{2}(\nabla+\nabla^{T})\overrightarrow{u},\end{equation}
-
-\end_inset
-
-so in this case our integral equation becomes
-\begin_inset Formula \begin{multline}
-\sum_{elements}(\int_{V^{e}}\frac{1}{4}(\nabla+\nabla^{T})\underline{N}:C\cdot(\nabla+\nabla^{T})\underline{N}\cdot\overrightarrow{u^{e}})\, dV+\int_{V^{e}}\rho\underline{N}\cdot\underline{N}\cdot\frac{\partial^{2}\overrightarrow{u^{e}}}{\partial t^{2}}\: dV-\int_{V^{e}}\underline{N}\cdot\underline{N}\cdot\overrightarrow{f^{e}}\, dV\\
--\int_{S_{T}}\underline{N}\cdot\underline{N}\cdot\overrightarrow{T^{e}}\, dS)=0.\end{multline}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Implicit Time Integration of Quasi-Static Equation
-\end_layout
-
-\begin_layout Subsubsection
-Index Notation
-\end_layout
-
-\begin_layout Standard
-Equation ??? may be linear or nonlinear, and may be solved using Newton-Raphson
- methods.
- We write a simplified version as a function of the displacements at time
-
-\begin_inset Formula $n+1$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \begin{equation}
-\boldsymbol{P}\left(\boldsymbol{u}_{n+1}\right)-\boldsymbol{F}_{n+1}=\mathbf{0}\equiv\boldsymbol{R}_{n+1}\end{equation}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Using the central difference method to approximate the acceleration (and
- velocity),
-\begin_inset Formula \begin{gather}
-\ddot{u}_{i}=\frac{1}{\Delta t^{2}}\left(u_{i}(t+\Delta t)-2u_{i}(t)+u_{i}(t-\Delta t)\right)\\
-\dot{u}_{i}=\frac{1}{2\Delta t}\left(u_{i}(t+\Delta t)-u_{i}(t-\Delta t)\right)\end{gather}
-
-\end_inset
-
-we have
-\begin_inset Formula \begin{multline}
-\sum_{elements}(\int_{V^{e}}\frac{1}{2}\sigma_{ij}(N_{,j}^{p}+N_{,i}^{p})\, dV+\int_{V^{e}}\rho N_{}^{p}\sum_{q}N_{}^{q}(u_{i}^{q}(t+\Delta t)-2u_{i}^{q}(t)+u_{i}^{q}(t-\Delta t))\: dV\\
--\int_{V^{e}}N^{p}f_{i}(t)\: dV-\int_{S_{T}}N^{p}T_{i}\: dS)=0,\end{multline}
-
-\end_inset
-
-which we can expand into
-\begin_inset Formula \begin{multline}
-\sum_{elements}({\frac{1}{\Delta t^{2}}\int}_{V^{e}}\rho N^{p}\sum_{q}N^{q}u_{i}^{q}(t+\Delta t)\, dV-\frac{2}{\Delta t^{2}}\int_{V^{e}}\rho N^{p}\sum_{q}N^{q}u_{i}^{q}(t)\, dV\\
-+\frac{1}{\Delta t^{2}}\int_{V^{e}}\rho N^{p}\sum_{q}N^{q}u_{i}^{q}(t-\Delta t)\, dV+{\frac{1}{2}\int}_{V^{e}}\sigma_{ij}(t)(N_{,j}^{p}+N_{,i}^{p})\: dV\\
--\int_{V^{e}}N^{p}f_{i}(t)\: dV-\int_{S_{T}}N^{p}T_{i}\: dS)=0\end{multline}
-
-\end_inset
-
-for the
-\begin_inset Formula $i$
-\end_inset
-
-th component associated with basis function
-\begin_inset Formula $N^{p}$
-\end_inset
-
-.
- Isolating the term containing
-\begin_inset Formula $u_{i}^{q}(t+\Delta t)$
-\end_inset
-
- yields
-\begin_inset Formula \begin{multline*}
-\frac{1}{\Delta t^{2}}\sum_{elements}\left(\int_{V^{e}}\rho N^{p}\sum_{q}N^{q}u_{i}^{q}(t+\Delta t)\, dV\right)=\frac{2}{\Delta t^{2}}\sum_{elements}\left(\int_{V^{e}}\rho N^{p}\sum_{q}N^{q}u_{i}^{q}(t)\, dV\right)\\
--\frac{1}{\Delta t^{2}}\sum_{elements}\left(\int_{V^{e}}\rho N^{p}\sum_{q}N^{q}u_{i}^{q}(t-\Delta t)\, dV\right)-\frac{1}{2}\sum_{elements}\left(\int_{V^{e}}\sigma_{ij}(t)(N_{,j}^{p}+N_{,i}^{p})\: dV\right)\\
-+\sum_{elements}\left(\int_{V^{e}}N^{p}f_{i}(t)\: dV\right)+\sum_{elements}\left(\int_{S_{T}}N^{p}T_{i}\: dS\right).\end{multline*}
-
-\end_inset
-
-We can rewrite the left-hand-side as a matrix-vector product where the vector
- is the displacement field at time
-\begin_inset Formula $t+\Delta t$
-\end_inset
-
- and the element mass matrix is given by
-\begin_inset Formula \[
-M_{ij}^{pq}=\delta_{ij}\int_{V^{e}}\rho N^{p}N^{q}\, dV,\]
-
-\end_inset
-
-where
-\begin_inset Formula $M_{ij}^{pq}$
-\end_inset
-
- is a
-\begin_inset Formula $pn$
-\end_inset
-
- by
-\begin_inset Formula $qn$
-\end_inset
-
- matrix (
-\begin_inset Formula $n$
-\end_inset
-
- is the dimension of the vector space),
-\begin_inset Formula $p$
-\end_inset
-
- and
-\begin_inset Formula $q$
-\end_inset
-
- refer to the basis functions and
-\begin_inset Formula $i$
-\end_inset
-
- and
-\begin_inset Formula $j$
-\end_inset
-
- are vector space components.
-
-\end_layout
-
-\begin_layout Subsubsection
-Vector Notation
-\end_layout
-
-\begin_layout Standard
-Using the central difference method to approximate the acceleration (and
- velocity),
-\begin_inset Formula \begin{gather}
-\frac{\partial^{2}\overrightarrow{u}(t)}{\partial t^{2}}=\frac{1}{\Delta t^{2}}\left(\overrightarrow{u}(t+\Delta t)-2\overrightarrow{u}(t)+\overrightarrow{u}(t-\Delta t)\right)\\
-\frac{\partial\overrightarrow{u}(t)}{\partial t}=\frac{1}{2\Delta t}\left(\overrightarrow{u}(t+\Delta t)-\overrightarrow{u}(t-\Delta t)\right)\end{gather}
-
-\end_inset
-
-
-\end_layout
-
-\end_body
-\end_document
Modified: short/3D/PyLith/trunk/doc/userguide/runpylith/runpylith.lyx
===================================================================
--- short/3D/PyLith/trunk/doc/userguide/runpylith/runpylith.lyx 2010-06-03 22:31:49 UTC (rev 16882)
+++ short/3D/PyLith/trunk/doc/userguide/runpylith/runpylith.lyx 2010-06-03 23:15:58 UTC (rev 16883)
@@ -1467,6 +1467,12 @@
\end_layout
\begin_layout Subsection
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:petsc:options"
+
+\end_inset
+
PETSc Settings (
\family typewriter
petsc
@@ -1475,7 +1481,8 @@
\end_layout
\begin_layout Standard
-PyLith relies on PETSc for the linear algebra computations.
+PyLith relies on PETSc for the linear algebra computations, including linear
+ Krylov subspace solvers and nonlinear solvers.
PETSc options can be set in
\family typewriter
.cfg
@@ -1497,6 +1504,16 @@
\end_inset
.
+ PETSc options are used to control the selection and settings for the solvers
+ underlying the SolverLinear and SolverNonlinear objects discussed in Section
+
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:solvers"
+
+\end_inset
+
+.
In many quasi-static or dynamic elasticity simulations, runtime can be
reduced by replacing the Jacobi preconditioner with Additive Schwartz with
Gram-Schmidt orthogonalization (see Table
@@ -2110,14 +2127,27 @@
facility to the general-problem.
The formulation specifies the time-stepping formulation to integrate the
elasticity equation.
- Implicit time stepping should be used for quasi-static problems, whereas
- explicit time stepping should be used for dynamic problems which include
+ Implicit time stepping is used for quasi-static problems and neglects interial
+ terms (see Section
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:elasticity:integral:quasistatic"
+
+\end_inset
+
+), whereas explicit time stepping is used for dynamic problems and includes
inertial terms.
- Two options for explicit time-stepping are available: using a full, consistent
- Jacobian matrix and a lumped Jacobian matrix.
+ There are two general options for explicit time-stepping: using a full,
+ consistent Jacobian matrix and a lumped Jacobian matrix.
In the lumped Jacobian formulation, the Jacobian is a diagonal matrix and
stored as a vector, and PyLith employs an optimized built-in solver rather
than a PETSc solver.
+ PyLith automatically switches to using the optimized built-in when the
+ lumped formulation is chosen.
+ PyLith also provides two optimized time-stepping options, one for single
+ point quadrature in linear triangular cells and one for single point quadrature
+ in linear tetrahedral cells.
+
\end_layout
\begin_layout Standard
@@ -2159,6 +2189,33 @@
formulation = pylith.problems.ExplicitLumped
\end_layout
+\begin_layout Standard
+An example of setting the formulation facility to the lumped explicit time
+ stepping component optimized for linear triangular cells with one point
+ quadrature is:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent]
+\end_layout
+
+\begin_layout LyX-Code
+formulation = pylith.problems.ExplicitLumpedTri3
+\end_layout
+
+\begin_layout Standard
+An example of setting the formulation facility to the lumped explicit time
+ stepping component optimized for linear tetrehedral cells is:
+\end_layout
+
+\begin_layout LyX-Code
+[pylithapp.timedependent]
+\end_layout
+
+\begin_layout LyX-Code
+formulation = pylith.problems.ExplicitLumpedTet4
+\end_layout
+
\begin_layout Subsection
Time-Stepping Formulation
\end_layout
@@ -2174,7 +2231,7 @@
\end_layout
\begin_layout Description
-solver Type of solver to use (default is linear).
+solver Type of solver to use (default is SolverLinear).
\end_layout
\begin_layout Description
@@ -2241,6 +2298,48 @@
\end_layout
\begin_layout Subsection
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:solvers"
+
+\end_inset
+
+Solvers
+\end_layout
+
+\begin_layout Standard
+PyLith supports three types of solvers.
+ The linear solver, SolverLinear, corresponds to the PETSc KSP solver and
+ is used in linear problems with linear elastic and viscoelastic bulk constituti
+ve models and kinematic fault ruptures.
+ The nonlinear solver, SolverNonlinear, corresponds to the PETSc SNES solver
+ and is used in nonlinear problems with nonlinear viscoelastic or elastoplastic
+ bulk constitutive models or dynamic fault ruptures.
+ The lumped solver (SolverLumped) is a specialized solver used with the
+ lumped system Jacobian matrix.
+ The options for the PETSc KSP and SNES solvers are set via the top-level
+ PETSc options (see Section
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:petsc:options"
+
+\end_inset
+
+ and the PETSc documentation
+\begin_inset Flex URL
+status collapsed
+
+\begin_layout Plain Layout
+
+www.mcs.anl.gov/petsc/petsc-as/documentation/index.html
+\end_layout
+
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Subsection
Time Stepping
\end_layout
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