Related papers: Automatic implementation of material laws: Jacobia…
This article aims to demonstrate and discuss the applications of automatic differentiation (AD) for finding derivatives in PDE-constrained optimization problems and Jacobians in non-linear finite element analysis. The main idea is to…
The efficient computation of Jacobians represents a fundamental challenge in computational science and engineering. Large-scale modular numerical simulation programs can be regarded as sequences of evaluations of in our case differentiable…
Most nonlinear partial differential equation (PDE) solvers require the Jacobian matrix associated to the differential operator. In PETSc, this is typically achieved by either an analytic derivation or numerical approximation method such as…
In this pedagogical article, we present a simple direct matrix method for analytically computing the Jacobian of nonlinear algebraic equations that arise from the discretization of nonlinear integro-differential equations. The method is…
We introduce a new set of algorithms to compute Jacobi matrices associated with measures generated by infinite systems of iterated functions. We demonstrate their relevance in the study of theoretical problems, such as the continuity of…
We describe efficient differentiation methods for computing Jacobians and gradients of a large class of matrix functions including the matrix logarithm $\log(A)$ and $p$-th roots $A^{\frac{1}{p}}$. We exploit contour integrals and conformal…
We examine aspects of the computation of finite element matrices and vectors which are made possible by automated code generation. Given a variational form in a syntax which resembles standard mathematical notation, the low-level computer…
The finite element method can be viewed as a machine that automates the discretization of differential equations, taking as input a variational problem, a finite element and a mesh, and producing as output a system of discrete equations.…
This contribution presents an improved low-order 3D finite element formulation with hourglass stabilization using automatic differentiation (AD). Here, the former Q1STc formulation is enhanced by an approximation-free computation of the…
A novel nonlinear formulation of the finite element and Galerkin methods is presented here, which leads to the Hadamard product expression of the resultant nonlinear algebraic analogue. The presented formulation attains the advantages of…
We propose an efficient finite-element analysis of the vector wave equation in a class of relatively general curved polygons. The proposed method is suitable for an accurate and efficient calculation of the propagation constants of…
The efficient computation of Jacobians represents a fundamental challenge in computational science and engineering. Large-scale modular numerical simulation programs can be regarded as sequences of evaluations of in our case differentiable…
Fractional-order differentiation has many characteristics different from integer-order differentiation. These characteristics can be applied to the optimization algorithms of artificial neural networks to obtain better results. However, due…
Code generation based software platforms, such as Firedrake, have become popular tools for developing complicated finite element discretisations of partial differential equations. We extended the code generation infrastructure in Firedrake…
We describe here a library aimed at automating the solution of partial differential equations using the finite element method. By employing novel techniques for automated code generation, the library combines a high level of expressiveness…
In order to accelerate implementation of hyperelastic materials for finite element analysis, we developed an automatic numerical algorithm that only requires the strain energy function. This saves the effort on analytical derivation and…
The finite element method is a well-established method for the numerical solution of partial differential equations (PDEs), both linear and nonlinear. However, the repeated reassemblage of finite element matrices for nonlinear PDEs is…
By using the Hadamard matrix product concept, this paper introduces two generalized matrix formulation forms of numerical analogue of nonlinear differential operators. The SJT matrix-vector product approach is found to be a simple,…
Jacobi-type iterative algorithms for the eigenvalue decomposition, singular value decomposition, and Takagi factorization of complex matrices are presented. They are implemented as compact Fortran 77 subroutines in a freely available…
Broyden's method is a general method commonly used for nonlinear systems of equations, when very little information is available about the problem. We develop an approach based on Broyden's method for nonlinear eigenvalue problems. Our…