50 research outputs found
An adaptive finite element method for laser surface hardening of steel problem
ACMAC’s PrePrint Repository aim is to enable open access to the scholarly output of ACMAC
A Nonconforming Finite Element Approximation for the von Karman Equations
In this paper, a nonconforming finite element method has been proposed and
analyzed for the von Karman equations that describe bending of thin elastic
plates. Optimal order error estimates in broken energy and norms are
derived under minimal regularity assumptions. Numerical results that justify
the theoretical results are presented.Comment: The paper is submitted to an international journa
A priori error estimates for the optimal control of laser surface hardening of steel
A priori error estimates for the optimal control of laser surface hardening of stee
Numerical analysis for the pure Neumann control problem using the gradient discretisation method
The article discusses the gradient discretisation method (GDM) for
distributed optimal control problems governed by diffusion equation with pure
Neumann boundary condition. Using the GDM framework enables to develop an
analysis that directly applies to a wide range of numerical schemes, from
conforming and non-conforming finite elements, to mixed finite elements, to
finite volumes and mimetic finite differences methods. Optimal order error
estimates for state, adjoint and control variables for low order schemes are
derived under standard regularity assumptions. A novel projection relation
between the optimal control and the adjoint variable allows the proof of a
super-convergence result for post-processed control. Numerical experiments
performed using a modified active set strategy algorithm for conforming,
nonconforming and mimetic finite difference methods confirm the theoretical
rates of convergence
Error estimates for the numerical approximation of a distributed optimal control problem governed by the von K\'arm\'an equations
In this paper, we discuss the numerical approximation of a distributed
optimal control problem governed by the von Karman equations, defined in
polygonal domains with point-wise control constraints. Conforming finite
elements are employed to discretize the state and adjoint variables. The
control is discretized using piece-wise constant approximations. A priori error
estimates are derived for the state, adjoint and control variables under
minimal regularity assumptions on the exact solution. Numerical results that
justify the theoretical results are presented
Convergence of an adaptive mixed finite element method for general second order linear elliptic problems
The convergence of an adaptive mixed finite element method for general second
order linear elliptic problems defined on simply connected bounded polygonal
domains is analyzed in this paper. The main difficulties in the analysis are
posed by the non-symmetric and indefinite form of the problem along with the
lack of the orthogonality property in mixed finite element methods. The
important tools in the analysis are a posteriori error estimators,
quasi-orthogonality property and quasi-discrete reliability established using
representation formula for the lowest-order Raviart-Thomas solution in terms of
the Crouzeix-Raviart solution of the problem. An adaptive marking in each step
for the local refinement is based on the edge residual and volume residual
terms of the a posteriori estimator. Numerical experiments confirm the
theoretical analysis.Comment: 24 pages, 8 figure
Dual weighted residual method for laser surface hardening of steel problem
Abstract. The main focus of this article is on the development of Adaptive Finite Element Method (AFEM) for the optimal control problem of laser surface hardening of steel governed by a dynamical system consisting of a semi-linear parabolic equation and an ordinary differential equation using Dual Weighted Residual Method (DWR). A posteriori error estimators using DWR method have been developed when a continuous piecewise linear discretization has been used for the finite element approximation of space variables and a discontinuous Galerkin method has been used for time and control discretizations. Further numerical results obtained are presented are compared with residual method numerical results. Key Words. Laser surface of steel problem, Adaptive finite element methods, Dual weighted residual methods, a posteriori error estimates. 1