Daniel Arndt,
Wolfgang Bangerth,
Denis Davydov,
- Graham Harper
+ Graham Harper,
Timo Heister,
Luca Heltai,
Martin Kronbichler,
\author[2]{Wolfgang~Bangerth}
\affil[2]{Department of Mathematics, Colorado State University, Fort
Collins, CO 80523-1874, USA.
- {\texttt{bangerth@colostate.edu},
+ {\texttt{bangerth@colostate.edu},
\texttt{harper@math.colostate.edu}}}
\author[3]{Denis~Davydov}
\label{subsec:ad}
In the previous release, numerous classes that are used to assemble linear systems
-and right hand sides, as well those used to define constitutive laws, were given
-full support for ``white-listed'' automatically differentiable (AD) number types
+and right hand sides, as well those used to define constitutive laws, were given
+full support for ``white-listed'' automatically differentiable (AD) number types
from the ADOL-C and Sacado libraries. This included the classes that
represent the local contributions of one cell to the global linear
system (i.e., \texttt{FullMatrix} and \texttt{Vector}) as well as the
In the first context, the finite element degrees of freedom are considered the
independent variables. From these primitives, the \texttt{EnergyFunctional} helper
-class in the namespace \texttt{Differentiation::AD} may be used to compute both the
-residual and its linearization by directly defining the contribution to the
-(twice differentiated) scalar total energy functional from each cell. Similarly,
-the \texttt{ResidualLinearization} class requires the (once-differentiated) finite
-element residual to be defined on a per cell basis, and this contribution is
+class in the namespace \texttt{Differentiation::AD} may be used to compute both the
+residual and its linearization by directly defining the contribution to the
+(twice differentiated) scalar total energy functional from each cell. Similarly,
+the \texttt{ResidualLinearization} class requires the (once-differentiated) finite
+element residual to be defined on a per cell basis, and this contribution is
automatically linearized.
The second context aims directly at constitutive model formulations, and serves to
compute the directional derivatives of components of (multi-field) constitutive laws
-with respect to the scalar, vector, tensor and symmetric tensor fields in terms
-of which they are parameterized. The \texttt{ScalarFunction} class may be used to
-define a scalar function (such as strain energy function) that may be twice
+with respect to the scalar, vector, tensor and symmetric tensor fields in terms
+of which they are parameterized. The \texttt{ScalarFunction} class may be used to
+define a scalar function (such as strain energy function) that may be twice
differentiated, while the \texttt{VectorFunction} may be used to define a vector
function (such as a set of kinematic fields) that may be differentiated once.
-Since the total derivatives of all components are computed at once, these two helper
-classes provide an interface to retrieve each sub-component of the gradient and
-Hessian (for a \texttt{ScalarFunction}) or values and Jacobian (for a
+Since the total derivatives of all components are computed at once, these two helper
+classes provide an interface to retrieve each sub-component of the gradient and
+Hessian (for a \texttt{ScalarFunction}) or values and Jacobian (for a
\texttt{VectorFunction}).
Although these aforementioned helper classes have been documented with a specific
use in mind, they remain generic and may (with a reinterpretation of the meaning of
the independent and dependent variables) be used for other purposes as well.
-Furthermore, through the implementation of \texttt{TapedDrivers} and
-\texttt{TapelessDrivers} classes that interface with the active AD library, the
-generic helper classes hide library-dependent implementational details and facilitate
-switching between the supported libraries and AD number types based on the
+Furthermore, through the implementation of \texttt{TapedDrivers} and
+\texttt{TapelessDrivers} classes that interface with the active AD library, the
+generic helper classes hide library-dependent implementational details and facilitate
+switching between the supported libraries and AD number types based on the
user's requirements.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Dedicated support for symbolic algebra, including algebra differentiation}
\label{subsec:sd}
To complement the automatic differentiation features in \dealii{}, this release sees
-the first step towards integrating and supporting a highly performant computer algebra
+the first step towards integrating and supporting a highly performant computer algebra
system (CAS) via the SymEngine library.
\marginpar{Could use a reference to SymEngine}
The \texttt{Expression} class in the namespace \texttt{Differentiation::SD} interfaces
-to SymEngine and forms the basis of symbolic computations, offering a full set of
-overloaded operators and a C++ style interface. This class offers the following
+to SymEngine and forms the basis of symbolic computations, offering a full set of
+overloaded operators and a C++ style interface. This class offers the following
basic functionality:
\begin{itemize}
\item symbolic variable definition,
\item substitution (partial and complete), and
\item serialization.
\end{itemize}
-\dealii{} now also provides an extensive set of math operations, with a syntax mimicking that used
-in the C++ standard library. Using the \texttt{Expression} class as a basis, we
+\dealii{} now also provides an extensive set of math operations, with a syntax mimicking that used
+in the C++ standard library. Using the \texttt{Expression} class as a basis, we
have developed a set of functions that can be used to create \dealii{} \texttt{Tensor}s
and \texttt{SymmetricTensor}s of symbolic variables and symbolic functions. This
gives full symbolic tensor algebra support using the pre-existing \texttt{Tensor}
with respect to other scalar expressions, tensor and symmetric tensors of expressions;
\item create symbolic substitution maps;
\item resolve explicit dependencies between expressions; and
-\item perform scalar and tensor valued substitution (including conversion from symbolic to
+\item perform scalar and tensor valued substitution (including conversion from symbolic to
real-valued scalars and tensors).
\end{itemize}
-In the next release we expect to implement classes to assist in performing assembly operations
+In the next release we expect to implement classes to assist in performing assembly operations
in the same spirit as that which has been done in the \texttt{Differentiation::AD} namespace,
although in a fully symbolic manner.
We will also address performance issues of the \texttt{Expression} class by leveraging
-the optimization capabilities of SymEngine, including common subexpression elimination (CSE),
+the optimization capabilities of SymEngine, including common subexpression elimination (CSE),
as well as by generating high performance code-paths to evaluate these expressions through the
-use of a custom-generated \texttt{std::function} (so-called ``lambda'' optimization) or by
+use of a custom-generated \texttt{std::function} (so-called ``lambda'' optimization) or by
compiling expressions using the LLVM JIT compiler.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{ljungkvist2017}. With this addition, both Dirichlet
boundary conditions and the constraints arising from adaptively
refined meshes can be imposed within the matrix-free framework. The only restriction is that for two-dimensional meshes the
- finite element degree must be odd.
+ finite element degree must be odd.
\marginpar{Can we explain this restriction? It seems odd (pun intended)...}
\marginpar{BT: No, we cannot. I have spent at least two full weeks trying to
understand why it doesn't work, i.e., gives the wrong result, but I don't know. We decided at the time that
projects / release-papers.git / commitdiff