Boundary-fitted coordinate systems for numerical solution of partial differential equations—A review

SUMMARY Given a list of points defining a domain boundary, a three-stage process is often used to triangulate a domain. First, an appropriate distribution of interior points is generated. Next the points are connected to form triangles. And, finally, the connectivity data are used to reposition the interior points using the Laplacian smoothing technique, thereby usually improving the shapes of the triangles. This paper describes a new technique for mesh improvement-adjusting the connection structure during the second stage of this process. The new scheme, which we call mesh relaxation, consists of a procedure for iteratively making the mesh topology more regular by edge swapping. For each interior edge, a relaxation index is computed that depends on the degrees of its end points and adjacent points. Any edge for which this index exceeds a prescribed threshold will be swapped, i.e. replaced by a new edge connecting the adjacent points of the original edge. After all edge swaps are completed, Laplacian smoothing is applied to the mesh. Examples show that, when the mesh point density varies smoothly and due care is taken in the vicinity of the boundary, mesh relaxation can dramatically increase the regularity of the mesh and produce improved triangle shapes.

Mesh relaxation: A new technique for improving triangulations

A new type of ensemble filter is proposed, which combines an ensemble Kalman filter (EnKF) with the ideas of morphing and registration from image processing. This results in filters suitable for nonlinear problems whose solutions exhibit moving coherent features, such as thin interfaces in wildfire modeling. The ensemble members are represented as the composition of one common state with a spatial transformation, called registration mapping, plus a residual. A fully automatic registration method is used that requires only gridded data, so the features in the model state do not need to be identified by the user. The morphing EnKF operates on a transformed state consisting of the registration mapping and the residual. Essentially, the morphing EnKF uses intermediate states obtained by morphing instead of linear combinations of the states.

Morphing Ensemble Kalman Filters

We present a technique to extend any Jacobian based quality measure for linear elements to high-order isoparametric planar triangles of any interpolation degree. The extended quality measure is obtained as the inverse of the distortion of the high-order element with respect to an ideal element. To measure the high-order distortion, we integrate on the curved element the inverse of the Jacobian based quality measure. Thus, we can proof that if the Jacobian based quality is invariant under a particular affine mapping, then the resulting quality measure is also invariant under that mapping. In addition, we check that the quality measure detects non-valid and low-quality high-order elements. Finally, we present and test an approach to generate curved meshes by minimizing the high-order distortion measure of the elements.

Defining Quality Measures for High-Order Planar Triangles and Curved Mesh Generation

A new type of ensemble filter is proposed, which combines an ensemble Kalman filter (EnKF) with the ideas of morphing and registration from image processing. This results in filters suitable for non-linear problems whose solutions exhibit moving coherent features, such as thin interfaces in wildfire modelling. The ensemble members are represented as the composition of one common state with a spatial transformation, called registration mapping, plus a residual. A fully automatic registration method is used that requires only gridded data, so the features in the model state do not need to be identified by the user. The morphing EnKF operates on a transformed state consisting of the registration mapping and the residual. Essentially, the morphing EnKF uses intermediate states obtained by morphing instead of linear combinations of the states.

/pdf/morphing-ensemble-kalman-filters-40api97psz.pdf

Morphing ensemble Kalman filters

This paper contains sufficient conditions under which a map whose domain is a compact set is a bijection onto a given set. Relative to certain isoparametric flnite element maps, one set of conditions involves the nonvanishing of the Jacobian; another the notion of overspill. An algorithm based on elimination is given for the numerical inversion of these maps.

/pdf/some-results-on-the-global-inversion-of-bilinear-and-57fh1n7vj9.pdf

Some results on the global inversion of bilinear and quadratic isoparametric finite element transformations

A system of algorithms is presented for the computer simulation of confined unsteady flows of a compressible fluid. The methods are valid for a wide range of time scales and are applied to simulate wind tunnel acoustics, flow over a ramp and flow past aircraft cavities.

A.E. Frey

Papers

Some results on the global inversion of bilinear and quadratic isoparametric finite element transformations

Numerical simulation of confined unsteady aerodynamical flows

An application of computer graphics to three dimensional finite element analyses

Numerical integration and the constant strain condition