Adaptive finite element computational fluid dynamics using an anisotropic error estimator
Citations
461 citations
Cites methods from "Adaptive finite element computation..."
...This is based on predicting good element sizes for the pore pressures using a Hessian (curvature)-based error estimator (Almeida et al., 2000), together with element sizes that directly minimise the bounds gap (as described in the section ‘Adaptive mesh refinement’)....
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...Using the nodal pore pressure field and the Hessian-based error estimator of Almeida et al. (2000), compute the 1 s p p ( ) [1 tanh( )] (1)2 α...
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...This is based on predicting good element sizes for the pore pressures using a Hessian (curvature)-based error estimator (Almeida et al., 2000), together with element sizes that directly minimise the bounds gap (as described in the section ‘Adaptive mesh refinement’). Where the element sizes predicted by these two separate approaches differ, the hybrid scheme simply chooses the smallest one. Details of the Hessian-based scheme for selecting element sizes, in the context of lower-bound limit analysis, can be found in Lyamin et al. (2005b). Exactly the same approach is used here, with the ‘isotropic’ form of the method being implemented, which omits element ‘stretching’....
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202 citations
Cites background or methods from "Adaptive finite element computation..."
...g 0 has been normalized to interval [0, 1] with 0 for flat tetrahedron and 1 for regular tetrahedron in transformed space....
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...The current mesh adaptivity techniques can be decomposed into three categories: • Adaptive re-meshing methods [36,20,1,33]; • Element subdivision methods [2,3,5,7,24,29,30,39,41]; • Fixed order mesh modification procedures [12,13,17,15,10,22,35]....
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...10(a) gives the initial mesh covering two-dimensional domain [0, 1] · [0, 1] with origin at its left-down corner....
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174 citations
118 citations
Cites background or methods from "Adaptive finite element computation..."
...Numerical Hessians in elements crossing the discontinuity are highly ill conditioned and cannot be used: across a shock of direction n, 2u/ n2 = n ·H ·n changes sign (see Figure 1)....
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...Therefore, the present work employs the reconstruction procedure of Section 3.2 to evaluate the Hessian matrix in the portions of the domains where the exact solution is assumed to be smooth....
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...One approach used with C0 finite element basis is the construction of a ‘recovered’ Hessian [3] using patchwise projection procedures in a manner similar to that used to define the popular Zienkiewicz– Zhu error estimators [27]....
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...They are to regenerate the mesh against that mesh size field [3, 4, 6, 12], or to perform appropriate local mesh modifications to match the desired mesh size field [9–11, 14]....
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...Therefore, anisotropic adaptive procedures employ the full set of second order derivatives (Hessian matrix) [4, 6, 12] or examine derivatives in the direction of specific mesh entities (typically edges) [9, 14] to obtain directional information on the desired mesh layout....
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112 citations
Additional excerpts
...In other words, we are not concerned with the discretization issue of th problem: we just assume that we have some accurate enough approximations of z, r and w to compute the quantitiesGK(z) in (16) andGK(r), L i,j K ( w) in (25), respectively (see also [1])....
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References
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"Adaptive finite element computation..." refers background in this paper
...The last optimal mesh has 4813 elements and sT ∈ [1; 4, 700]....
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...In particular Figure 11 shows the first uniform mesh (442 elements) and the last adapted mesh (step 4, 2820 elements, sT ∈ [1; 630])....
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...Moreover, in these meshes we have sT ∈ [1; 1, 724]....
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...In particular, Figure 9 shows the first uniform mesh (394 elements) and the last adapted mesh (step 3, 1466 elements and sT ∈ [1; 800])....
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