Solving Biharmonic Equations with Tri-Cubic C1 Splines on Unstructured Hex Meshes
Abstract
:1. Introduction
- a (fourth-order) convergence rate for Poisson’s equation on irregular box-complexes,
- i.e., the error between the computed and the known exact solution in the norm decreases by a factor of under halving of the mesh interval h, by in the error, and in the norm.
- for the regular case and for elements on geometry.
- convergence rate of singular tri-3 splines on singular tri-3 spline geometry is less than on irregular box-complexes.
- in all cases enumerated above, tri-3 splines exhibit faster convergence than Catmull-Clark solids.
- Overview. After a brief literature review of trivariate smooth elements and the bivariate antecedents of tri-3 splines, Section 2 defines the tri-3 splines for unstructured box-complexes. The space is on regular local grids but is if initialized by knot insertion on a locally tensor-product grid. The space has zero first derivatives across irregularities but is after a change in variables and has linear-independent B-spline-like basis functions per box. Section 4 shows the numerical convergence for Poisson’s equation and compares the convergence to that of Catmull-Clark solids. Section 5 shows and discusses the convergence for the biharmonic equation and compares it to Catmull-Clark solids.
2. Smooth Trivariate Finite Elements
2.1. Singular Jet Collapse Constructions in Two Variables
2.2. Constructions in Three Variables
3. Tri-3 Splines on Unstructured Box-Complexes
- Notation and Indexing. Analogous to a simplicial complex, a box-complex (also known as a hex-mesh) in is a collection of d-dimensional boxes, , called d-boxes. Boxes of any dimension overlap only in complete lower-dimensional d-boxes. A 0-box is a vertex, a 1-box an edge, a 2-box a quadrilateral, and a 3-box is a quadrilateral-faced hexahedron. A box without a prefix is a 3-box.
- Irregularities. For , an interior d-box is regular if it is completely surrounded by boxes and for , all incident -boxes are regular. For example, for a vertex to be regular, all edges incident to it must be regular. In , a regular vertex () is surrounded by 8 boxes, a regular edge () by 4 boxes, and a regular quadrilateral face () by 2 boxes. Interior faces are always regular since they are shared by exactly boxes.
- Polynomial pieces, corner inner, and index-wise nearest coefficients. A tri-3 spline consists of polynomial pieces represented in tri-variate tensor-product Bernstein-Bézier (BB) form (see [51] or [52]):
- The tri-3 splines. The splines are constructed by the following Algorithm 1.
Algorithm 1: Construction of tri-3 splines. Input: box-complex with vertices and values .Output: Tri-3 splines- Initialize the inner BB-coefficients , of each tri-3 piece by B-spline to BB-form conversion (knot insertion). In regular regions, the splines are, therefore, initially .
- Set the inner BB-coefficients of faces, the edges, and finally, the vertex as the average of their index-wise nearest neighbors.
- For irregular boxes only, apply de Casteljau’s algorithm to split each tri-3 piece into pieces.
- For irregular sub-boxes apply the operator from Appendix A (cf. Section 6 of [11])
- The isogeometric approach. Let □ be a cube and
4. Solving Poisson’s Equation over Unstructured Hex Meshes
Comparison to Catmull-Clark Solids
5. Solving the Biharmonic Equation over Unstructured Hex Meshes
6. Conclusions and Future Work
Author Contributions
Funding
Conflicts of Interest
Appendix A. Projector P
- 1.
- Compute a best-fit linear map ℓ to the BB-coefficients , , e.g., by computing a vector ℓ as
- 2.
- For each irregular sub-box , compute the polyhedral intersection by solving the system
- 3.
- For each irregular sub-box , compute the BB-coefficients of the singular parameterization by
- 4.
- For each irregular sub-box , for ,For each inner BB-coefficient of a semi-regular edge of valence the bi-variate 2-neighborhood ‘orthogonal’ to the edge is transformed via , a jet collapse followed by projection. is applied to each coordinate of separately, generating defined byHere with the argument interpreted modulo .
- 5.
- For every face with BB-coefficients shared by an irregular sub-box and a (regular or irregular) sub-box , enforce regular continuity by averaging
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Youngquist, J.; Peters, J. Solving Biharmonic Equations with Tri-Cubic C1 Splines on Unstructured Hex Meshes. Axioms 2022, 11, 633. https://doi.org/10.3390/axioms11110633
Youngquist J, Peters J. Solving Biharmonic Equations with Tri-Cubic C1 Splines on Unstructured Hex Meshes. Axioms. 2022; 11(11):633. https://doi.org/10.3390/axioms11110633
Chicago/Turabian StyleYoungquist, Jeremy, and Jörg Peters. 2022. "Solving Biharmonic Equations with Tri-Cubic C1 Splines on Unstructured Hex Meshes" Axioms 11, no. 11: 633. https://doi.org/10.3390/axioms11110633