A finite element algorithm for high-lying eigenvalues with Neumann and Dirichlet boundary conditions

2014 ◽  
Author(s):  
G. Báez ◽  
R. A. Méndez-Sánchez ◽  
F. Leyvraz ◽  
T. H. Seligman
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Element Algorithm

Checking the Automated Construction of Finite Element Simulations From Dirichlet Boundary Conditions

2019 ◽  
Author(s):  
Kevin N. Chiu ◽  
Mark D. Fuge
Keyword(s):  
Finite Element ◽  
Topology Optimization ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Fem Simulation ◽  
Computational Design ◽  
Dirichlet Boundary Conditions ◽  
Governing Equations ◽  
Variational Form ◽  
Key Issues

Abstract From engineering analysis and topology optimization to generative design and machine learning, many modern computational design approaches require either large amounts of data or a method to generate that data. This paper addresses key issues with automatically generating such data through automating the construction of Finite Element Method (FEM) simulations from Dirichlet boundary conditions. Most past work on automating FEM assumes prior knowledge of the physics to be run or is limited to a small number of governing equations. In contrast, we propose three improvements to current methods of automating the FEM: (1) completeness labels that guarantee viability of a simulation under specific conditions, (2) type-based labels for solution fields that robustly generate and identify solution fields, and (3) type-based labels for variational forms of governing equations that map the three components of a simulation set — specifically, boundary conditions, solution fields, and a variational form — to each other to form a viable FEM simulation. We implement these improvements using the FEniCS library as an example case. We show that our improvements increase the percent of viable simulations that are run automatically from a given list of boundary conditions. This paper’s procedures ultimately allow for the automatic — i.e., fully computer-controlled — construction of FEM multi-physics simulations and data collection required to run data-driven models of physics phenomena or automate the exploration of topology optimization under many physics.

scholarly journals A Nitsche Finite Element Approach for Elliptic Problems with Discontinuous Dirichlet Boundary Conditions

2018 ◽  
Vol 18 (3) ◽  
pp. 373-381 ◽  
Author(s):  
Ramona Baumann ◽  
Thomas P. Wihler
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Mesh Size ◽  
Singular Part ◽  
Dirichlet Boundary Conditions ◽  
Diffusion Reaction ◽  
Discrete Solution ◽  
Finite Element Approach ◽  
Element Approach

AbstractWe present a numerical approximation method for linear elliptic diffusion-reaction problems with possibly discontinuous Dirichlet boundary conditions. The solution of such problems can be represented as a linear combination of explicitly known singular functions as well as of an {H^{2}}-regular part. The latter part is expressed in terms of an elliptic problem with regularized Dirichlet boundary conditions, and can be approximated by means of a Nitsche finite element approach. The discrete solution of the original problem is then defined by adding back the singular part of the exact solution to the Nitsche approximation. In this way, the discrete solution can be shown to converge of second order in the {L^{2}}-norm with respect to the mesh size.

scholarly journals MIXED FINITE ELEMENT APPROXIMATION OF THE VECTOR LAPLACIAN WITH DIRICHLET BOUNDARY CONDITIONS

2012 ◽  
Vol 22 (09) ◽  
pp. 1250024 ◽  
Author(s):  
DOUGLAS N. ARNOLD ◽  
RICHARD S. FALK ◽  
JAY GOPALAKRISHNAN
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Stokes Equations ◽  
Mixed Finite Element ◽  
Mixed Finite Elements ◽  
Mixed Finite Element Method ◽  
Two Dimensions ◽  
Dirichlet Boundary Conditions ◽  
Element Approximation

We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is introduced as a second unknown, is advantageous, and appropriate choices of mixed finite element spaces lead to a stable, optimally convergent discretization. However, the theory that leads to these conclusions does not apply to the case of Dirichlet boundary conditions, in which both components of the solution vanish on the boundary. We show, by computational example, that indeed such mixed finite elements do not perform optimally in this case, and we analyze the suboptimal convergence that does occur. As we indicate, these results have implications for the solution of the biharmonic equation and of the Stokes equations using a mixed formulation involving the vorticity.

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