TED-AJ03-353 A METHOD FOR REDUCING THE ERRORS IN FICTITIOUS DOMAIN METHODS APPLIED TO HEAT TRANSFER PROBLEMS

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概要

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• Fictitious domain methods reduce the complexity of meshing a complex domain by extending it into a simpler one that can be discretized easily e.g. a rectangle in two dimension. The original problem is preserved by augmenting a functional defined over the new extended domain with a term that enforces the original boundary conditions. A possible fictitious domain method, attractive for its simplicity, is based on using Lagrange multipliers distributed over the complex boundary. The locations of the multipliers do not necessarily coincide with the underlying mesh and may fall within elements. The presence of a multiplier within an element is associated with the introduction of the fictitious domain. This can introduce significant errors on the adjacent nodes when the extended solution involves a discontinuous derivative across the embedded boundary, e.g. enforcing a Dirichlet boundary condition within the extended domain. The errors are due to an inappropriate approximation of the local gradient at the location of a multiplier. This is illustrated in Figure A. A one-dimensional model problem is first used to study possible improvements. To reduce the errors, it is first possible to refine the mesh. It is observed that a uniform reduction in the element size changes the relative location of the Lagrange multiplier within an element and the errors are found to oscillate as the mesh is refined uniformly for a given extended domain. The errors are also functions of the relative locations of the Lagrange multipliers within the elements for a fixed element size. For fixed relative Lagrange multiplier positions, the Lagrange multiplier/fictitious domain methods (LM/FDM) result in a lower-order convergence rate in L_2 norm compared to a solution based on a body conforming mesh. The magnitudes of the errors are related to the jump of the local solution gradient and the relative location of the multiplier within the corresponding element. The errors are minimized as the jump of the local gradient disappears. A two-step predictor-corrector method is proposed to mitigate the errors. In this approach, instead of satisfying the original embedded Dirichlet boundary condition, the local boundary conditions are adjusted to values that reduced significantly the errors. These local values are called the fictitious constraints and are obtained based on the computed solution from the standard Lagrange multiplier/fictitious domain method. The reduction in the errors can be observed in Figure B. In this method, the solution is assumed to be linear near the embedded boundary. Computation of the errors shows that the solutions obtained with the fictitious constraints are not very sensitive to the jump of the local gradients and the relative location of the Lagrange multipliers within elements. A higher convergence rate for the L_2 norm is observed when the relative Lagrange multiplier position is fixed. This is observed by solving simple one- and two-dimensional problems.[figure]

• 一般社団法人日本機械学会の論文

著者

• Li Yanbing

  Department Of Mechanical Engineering Michigan State University

• Benard Andre

  Department Of Mechanical Engineering Michigan State University

• LI Yanbing

  Department of Endocrinology and Diabetes Center, The First Affiliated Hospital of Sun Yat-sen University

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• TED-AJ03-353 A METHOD FOR REDUCING THE ERRORS IN FICTITIOUS DOMAIN METHODS APPLIED TO HEAT TRANSFER PROBLEMS

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