Grid adaptive methods combined with means for automatic remeshing are applied to problems in shape optimal design of linearly elastic structures. The quantitative effect of element distortion near the design boundaries is identified in terms of interpolation error associated with the finite element discretization. The grid adaptation is itself formulated as a structural optimization problem, with an objective function that reflects the discretization error. A 'necessary condition' from this formulation provides the basis for a computational procedure to predict the modified grid.To avoid the sometimes drastic distortion of the PBM grid that might otherwise occur in conjunction with design change, remeshing must be performed at intermediate stages of the overall solution process. In order to produce results for the optimal shape design without interruption in this process, the computer program combines numerical grid generation and automatic remeshing with the grid adaptation and design change. Results for several shape design problems obtained with the use of grid adaptation are compared to computational results predicted from a fixed grid. Both 'r-' and 'hadaptation' are tested.
Grid adaptive methods combined with domain adaptation are discussed for two-dimensional seepage flow problems with free boundaries through porous media. Examples of grid and domain adaptive methods are presented to demonstrate several ways to predict grids and shapes of free boundaries using an iterative scheme. Finally, the combined adaptive methods are applied to obtain smooth non-oscillatory shape of a free boundary of seepage flow through non-homogeneous porous media. (1) 1 K e , , = K + -( l -~n ) & where j(, is the characteristic function of the flow domain R defined by x,= 1 if XER, x,=O if ~$ 0 and E is a regularity parameter which is expected to be a very small positive real number, for example, ~= ( 1 0 -~-l O -~) K , , ,~, ,and Kmin is the minimum of the K in the flow domain. Mathematically, this extended pressure head method was introduced by Ah4*' and was also studied by Brezis et aL6 Application of these mathematicians' work to solve free boundary
Grid adaptive methods combined with means for automatic remeshing are applied to problems in shape optimal design of linearly elastic structures. The quantitative effect of element distortion near the design boundaries is identified in terms of interpolation error associated with the finite element discretization. The grid adaptation is itself formulated as a structural optimization problem, with an objective function that reflects the discretization error. A 'necessary condition' from this formulation provides the basis for a computational procedure to predict the modified grid.To avoid the sometimes drastic distortion of the PBM grid that might otherwise occur in conjunction with design change, remeshing must be performed at intermediate stages of the overall solution process. In order to produce results for the optimal shape design without interruption in this process, the computer program combines numerical grid generation and automatic remeshing with the grid adaptation and design change. Results for several shape design problems obtained with the use of grid adaptation are compared to computational results predicted from a fixed grid. Both 'r-' and 'hadaptation' are tested.
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