8,087 research outputs found
Sparse Automatic Differentiation for Large-Scale Computations Using Abstract Elementary Algebra
Most numerical solvers and libraries nowadays are implemented to use
mathematical models created with language-specific built-in data types (e.g.
real in Fortran or double in C) and their respective elementary algebra
implementations. However, built-in elementary algebra typically has limited
functionality and often restricts flexibility of mathematical models and
analysis types that can be applied to those models. To overcome this
limitation, a number of domain-specific languages with more feature-rich
built-in data types have been proposed. In this paper, we argue that if
numerical libraries and solvers are designed to use abstract elementary algebra
rather than language-specific built-in algebra, modern mainstream languages can
be as effective as any domain-specific language. We illustrate our ideas using
the example of sparse Jacobian matrix computation. We implement an automatic
differentiation method that takes advantage of sparse system structures and is
straightforward to parallelize in MPI setting. Furthermore, we show that the
computational cost scales linearly with the size of the system.Comment: Submitted to ACM Transactions on Mathematical Softwar
Monolithic simulation of convection-coupled phase-change - verification and reproducibility
Phase interfaces in melting and solidification processes are strongly
affected by the presence of convection in the liquid. One way of modeling their
transient evolution is to couple an incompressible flow model to an energy
balance in enthalpy formulation. Two strong nonlinearities arise, which account
for the viscosity variation between phases and the latent heat of fusion at the
phase interface.
The resulting coupled system of PDE's can be solved by a single-domain
semi-phase-field, variable viscosity, finite element method with monolithic
system coupling and global Newton linearization. A robust computational model
for realistic phase-change regimes furthermore requires a flexible
implementation based on sophisticated mesh adaptivity. In this article, we
present first steps towards implementing such a computational model into a
simulation tool which we call Phaseflow.
Phaseflow utilizes the finite element software FEniCS, which includes a
dual-weighted residual method for goal-oriented adaptive mesh refinement.
Phaseflow is an open-source, dimension-independent implementation that, upon an
appropriate parameter choice, reduces to classical benchmark situations
including the lid-driven cavity and the Stefan problem. We present and discuss
numerical results for these, an octadecane PCM convection-coupled melting
benchmark, and a preliminary 3D convection-coupled melting example,
demonstrating the flexible implementation. Though being preliminary, the latter
is, to our knowledge, the first published 3D result for this method. In our
work, we especially emphasize reproducibility and provide an easy-to-use
portable software container using Docker.Comment: 20 pages, 8 figure
A Numerical Slow Manifold Approach to Model Reduction for Optimal Control of Multiple Time Scale ODE
Time scale separation is a natural property of many control systems that can
be ex- ploited, theoretically and numerically. We present a numerical scheme to
solve optimal control problems with considerable time scale separation that is
based on a model reduction approach that does not need the system to be
explicitly stated in singularly perturbed form. We present examples that
highlight the advantages and disadvantages of the method
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