Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods

Robust Preconditioners for the High-Contrast Elliptic Partial Differential Equations

In this thesis, we discuss a robust preconditioner (the AGKS preconditioner) for solving linear systems arising from approximations of partial differential equations (PDEs) with high-contrast coefficients. The problems considered here include the standard second and higher order elliptic PDEs such as high-contrast diffusion equation, Stokes\u27 equation and biharmonic-plate equation. The goal of this study is the development of robust and parallelizable preconditioners that can easily be integrated to treat large configurations. The construction of the preconditioner consists of two phases. The first one is an algebraic phase which partitions the degrees of freedom into high and low permeability regions which may be of arbitrary geometry. This yields a corresponding block partitioning of the stiffness matrix allowing us to use a formula for the action of its inverse involving the inverses of both the high permeability block and its Schur complement in the original matrix. Singular perturbation analysis plays a big role to analyze the structure of the required subblock inverses in the high contrast case which shows that for high enough contrast each of the subblock inverses can be approximated well by solving only systems with constant coefficients. The second phase involves an efficient multigrid approximation of this exact inverse. After applying singular perturbation theory to each of the sub-blocks, we obtain that inverses of each of the subblocks with high contrast entries can be approximated efficiently using geometric multigrid methods, and that this approximation is robust with respect to both the contrast and the mesh size. The result is a multigrid method for high contrast problems which is provably optimal to both contrast and mesh size. We demonstrate the advantageous properties of the AGKS preconditioner using experiments on model high-contrast problems. We examine its performance against multigrid method under varying discretizations of diffusion equation, Stokes equation and biharmonic-plate equation. Thus, we show that we accomplished a desirable preconditioning design goal by using the same family of preconditioners to solve the elliptic family of PDEs with varying discretizations

Extreme Learning Machine-Assisted Solution of Biharmonic Equations via Its Coupled Schemes

Obtaining the solutions of partial differential equations based on various machine learning methods has drawn more and more attention in the fields of scientific computation and engineering applications. In this work, we first propose a coupled Extreme Learning Machine (called CELM) method incorporated with the physical laws to solve a class of fourth-order biharmonic equations by reformulating it into two well-posed Poisson problems. In addition, some activation functions including tangent, gauss, sine, and trigonometric (sin+cos) functions are introduced to assess our CELM method. Notably, the sine and trigonometric functions demonstrate a remarkable ability to effectively minimize the approximation error of the CELM model. In the end, several numerical experiments are performed to study the initializing approaches for both the weights and biases of the hidden units in our CELM model and explore the required number of hidden units. Numerical results show the proposed CELM algorithm is high-precision and efficient to address the biharmonic equation in both regular and irregular domains

Advanced Computational Engineering

The finite element method is the established simulation tool for the numerical solution of partial differential equations in many engineering problems with many mathematical developments such as mixed finite element methods (FEMs) and other nonstandard FEMs like least-squares, nonconforming, and discontinuous Galerkin (dG) FEMs. Various aspects on this plus related topics ranging from order-reduction methods to isogeometric analysis has been discussed amongst the pariticpants form mathematics and engineering for a large range of applications

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal