Automated Computational Modeling
Doctoral thesis, 2006
This thesis is part of the FEniCS project of Automation of Computational Mathematical Modeling (ACMM) as the modern manifestation of the basic principle of science:
formulating mathematical equations (modeling) and solving equations (computation).
The vision of FEniCS is to set a new standard towards the goals of generality, efficiency, and simplicity, concerning mathematical methodology, implementation, and application.
ACMM includes the key steps of Automation of (a) discretization of differential equations,
(b) solution of discrete systems, (c) error control of discrete solutions, (d) optimization
and (e) modeling. FEniCS is based on adaptive finite element methods (FEM) .
This thesis presents the following examples of Automated Computational Modeling as
concrete realizations of ACMM with main focus on (a)-(c):
MG: Multi-adaptive Galerkin ODE-solver: This part concerns the automation of (a1) discretization in time by the MG implementation of the multi-adaptive ODE-solver mcG(q)/mdG(q) formulated in the thesis () by Anders Logg, based on Galerkin's method with continuous/discontinuous piecewise polynomial approximation in time of degree q with different time steps for different components, automati- cally determined by a posteriori error estimation. MG realizes automation of (a1) by Galerkin's method, (b) by fixed-point or Newton's method, and (c) by a posteriori error estimation using duality. MG is the first general multi-adaptive ODE-solver with automatic error control based on duality. MG has potentially a very vast range of applicability. MG may also be run in mono-adaptive form with the same time step for all components, eliminating the over-head required for multi-adaptivity. For bench-mark problems with different time scales, we demonstrate substantial performance gains with MG, as compared to mono-adaptive solvers. MG is joint work with Anders Logg.
Ko: Solid mechanics solver: This part includes the automation of (a2) discretization in space of a general continuum model for solid mechanics with elasto-visco-plastic materials and large displacements, rotations and deformations. When coupled with MG for time-discretization this gives the solid mechanics solver Ko realizing (a) and (b) with (c) in progress. Ko is based on an updated Lagrangian formulation where equilibrium and constitutive equations are expressed on the current deformed configuration. Ko is the first automated solid mechanics solver and the range of possible applications is very large. We show that the performance of Ko is comparable to that of a mass-spring solver, which is the industry standard for performance-intensive solid mechanics simulations . Ko demonstrates the general capability and potential of FEniCS.
DOLFIN as a PSE: We present the FEniCS tool chain, and in particular DOLFIN, as a general and automated problem solving environment (PSE). DOLFIN realizes the overall concept of automated computational modeling by taking a PDE in mathematical notation as input and automatically discretizing and computing the solution by the FEM with full efficiency, including automated time discretization of time dependent PDE with the ODE solver. This is joint work with Johan Hoffman, Anders Logg and Garth Wells.
Automated Modeling: We present a case study of automated modeling (e) in a model problem with a fast and a slow time scale. By resolving the fast time scale for a short period of time an effective coefficient is determined by optimization, which allows simulation of the slow time scale over long time. This is joint work with Claes Johnson and Anders Logg.