Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem

2020 ◽  
Vol 26 ◽  
pp. 78
Author(s):  
Thirupathi Gudi ◽  
Ramesh Ch. Sau
Keyword(s):  
Finite Element ◽  
Control Problem ◽  
Dirichlet Boundary ◽  
A Priori ◽  
Diffusion Problem ◽  
Discrete Control ◽  
Optimal Order ◽  
Control Constraints ◽  
Element Analysis ◽  
Dirichlet Boundary Control

We study an energy space-based approach for the Dirichlet boundary optimal control problem governed by the Laplace equation with control constraints. The optimality system results in a simplified Signorini type problem for control which is coupled with boundary value problems for state and costate variables. We propose a finite element based numerical method using the linear Lagrange finite element spaces with discrete control constraints at the Lagrange nodes. The analysis is presented in a combination for both the gradient and the L2 cost functional. A priori error estimates of optimal order in the energy norm is derived up to the regularity of the solution for both the cases. Theoretical results are illustrated by some numerical experiments.

scholarly journals Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions

2018 ◽  
Vol 52 (6) ◽  
pp. 2247-2282 ◽  
Author(s):  
Erik Burman ◽  
Peter Hansbo ◽  
Mats G. Larson ◽  
André Massing
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Stiffness Matrix ◽  
A Priori ◽  
Beltrami Operator ◽  
Optimal Order ◽  
Codimension One ◽  
Special Cases ◽  
Arbitrary Codimension ◽  
Background Mesh

We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in ℝd of arbitrary codimension. The method is based on using continuous piecewise linears on a background mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the background mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order a priori estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the background mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in ℝ3.

Error analysis of PML-FEM approximations for the Helmholtz equation in waveguides

2019 ◽  
Vol 53 (4) ◽  
pp. 1191-1222 ◽  
Author(s):  
Seungil Kim
Keyword(s):  
Finite Element ◽  
Helmholtz Equation ◽  
Piecewise Linear ◽  
A Priori ◽  
Approximate Solutions ◽  
Dirichlet Condition ◽  
Neumann Condition ◽  
Element Analysis ◽  
The Finite Element Analysis ◽  
Two Parameters

In this paper, we study finite element approximate solutions to the Helmholtz equation in waveguides by using a perfectly matched layer (PML). The PML is defined in terms of a piecewise linear coordinate stretching function with two parameters for absorbing propagating and evanescent components respectively, and truncated with a Neumann condition on an artificial boundary rather than a Dirichlet condition for cutoff modes that waveguides may allow. In the finite element analysis for the PML problem, we have to deal with two difficulties arising from the lack of full regularity of PML solutions and the anisotropic nature of the PML problem with, in particular, large PML damping parameters. Anisotropic finite element meshes in the PML regions depending on the damping parameters are used to handle anisotropy of the PML problem. As a main goal, we establish quasi-optimal a priori error estimates, that does not depend on anisotropy of the PML problem (when no cutoff mode is involved), including the exponentially convergent PML error with respect to the width and the strength of PML. The numerical experiments that confirm the convergence analysis will be presented.

Shape Optimization of Potentially Slender Structures

2008 ◽  
Author(s):  
Kavous Jorabchi ◽  
Joshua Danczyk ◽  
Krishnan Suresh
Keyword(s):  
Finite Element ◽  
Shape Optimization ◽  
Dimensional Reduction ◽  
A Priori ◽  
Computational Cost ◽  
Optimization Process ◽  
Element Analysis ◽  
Dimensional Reduction Method ◽  
Modern Engineering ◽  
Geometric Reduction

Shape optimization lies at the heart of modern engineering design. Through shape optimization, computers can, in theory, ‘synthesize’ engineering artifacts in a fully automated fashion. However, a serious limitation today is that the evolving geometry (during optimization) may become slender, i.e., beam or plate-like. Under such circumstances, modern 3-D computational methods, such as finite element analysis (FEA), will fail miserably, and so will the shape optimization process. Indeed, the recommended method for analyzing slender artifacts is to replace them with 1-D beams/ 2-D plates, prior to discretization and computational analysis, a process referred to as geometric dimensional reduction. Unfortunately explicit geometric reduction is impractical and hard to automate during optimization since one cannot predict a priori when an artifact will become slender. In this paper, we develop an implicit dimensional reduction method where the reduction is achieved through an algebraic process. The proposed method of reduction is computationally equivalent to explicit geometric reduction for comparable computational cost. However, the proposed method can be easily automated and integrated within a shape optimization process, and standard off-the-shelf 3-D finite element packages can be used to implement the proposed methodology.

Sign in / Sign up

Export Citation Format

Share Document