Extending the functionality of the general‐purpose finite element package SEPRAN by automatic differentiation

Abstract

SUMMARYFrom an abstract point of view, a numerical simulation implements a mathematical function that produces some output from some given input. Derivatives (or sensitivities) of the function's output with respect to its input can be obtained-free from truncation error-by using a technique called automatic di erentiation. Given a computer code in a high-level programming language like Fortran, C, or C++, automatic di erentiation generates another code capable of computing not only the original function but also its derivatives. Thus, the application of automatic di erentiation signiÿcantly extends the functionality of a simulation package. For instance, automatic di erentiation enables, in a completely mechanical fashion, the usage of derivative-based optimization algorithms where the evaluation of the objective function comprises some given large-scale engineering simulation. In this paper, the automatic di erentiation tool ADIFOR is used to transform the general-purpose ÿnite element package SEPRAN. In doing so, we automatically transform the given 400 000 lines of Fortran 77 into a new program consisting of 600 000 lines of Fortran 77. We compare our approach with a traditional approach based on numerical di erentiation and quantify its advantages in terms of accuracy and computational e ciency for a standard uid ow problem.

Practical shape optimization of a levitation device for single droplets

“…This is a non-trivial task, and the SEPRAN package is one of the biggest codes successfully differentiated to date. In previous studies (Bischof et al 2003e) we reported on the differentiation of the SEPRAN package using the ADIFOR tool, and we also demonstrated the correctness of the sensitivities for the problem of a levitated droplet (Bischof et al 2003d). …”

Automation of primal and sensitivity analysis of transient coupled problems

“…AD (see e.g. Griewank [10], Bartholomew-Biggs et al [2], Bischof et al [4]) is a method to evaluate the derivative of a function specified by a computer program and represents an alternative solution to the numerical differentiation as well as to the symbolic differentiation. The AD technique is based on the fact that every computer program executes a sequence of elementary operations with known derivatives, thus allowing the evaluation of exact derivatives via the chain rule for an arbitrary complex formulation.…”

“…The nodal unknowns form the vector of element unknowns p e = {u 1 , v 1 , u 2 , v 2 , u 3 , v 3 , u 4 , v 4 }. The hyperelastic strain energy density function W is formulated as a function of invariants of the right Cauchy strain tensor C = F T F and Lame constants λ and µ as follows…”