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Xiaodong Wei is currently a postdoctoral fellow in Mathematics in Computational Science and Engineering at EPFL. He earned his Ph.D. in Mechanical Engineering at Carnegie Mellon University. He also received M.S. and B.S. in Civil Engineering from The University of Tokyo and Tsinghua University, respectively. His research interests lie broadly in the interdisciplinary field of computational mechanics and computational geometry, as well as their applications in engineering design/optimization, computational neuroscience, and additive manufacturing. He has been organizing mini-symposia in international conferences to promote real-world applications of computational mechanics and geometry, and he has been awarded with several fellowships, such as Bertucci Graduate Fellowship, Liang Ji-Dian Graduate fellowship and CIT Dean's Fellowship.

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Quadratic serendipity element shape functions on general planar polygons

Juan Cao, Xiaodong Wei

This paper proposes a method for the construction of quadratic serendipity element (QSE) shape functions on planar convex and concave polygons. Existing approaches for constructing QSE shape functions are linear combinations of the pair-wise products of generalized barycentric coordinates with linear precision, restricted to the convex polygonal domain or resort to numerical optimization. We extend the construction to general polygons with no more than three collinear consecutive vertices. This is done by defining coefficients of the linear combination as the oriented area of triangles with vertices from the polygonal domain, which can be either convex or concave. The proposed shape functions possess linear to quadratic precision. We prove the interpolation error estimates for mean value coordinate-based QSE shape functions on convex and concave polygonal domains satisfying a set of geometric constraints for standard finite element analysis. We also tailor a polygonal mesh generation scheme that improves the uniformity and avoids short edges of Voronoi diagrams for their use in the QSE-based polygonal finite element computation. Numerical tests for the 2D Poisson equations on various domains are presented, demonstrating the optimal convergence rates in both the L-2-norm and the H-1-seminorm. 2022 Elsevier B.V. All rights reserved.

ELSEVIER SCIENCE SA2022

Robust numerical integration on curved polyhedra based on folded decompositions

Pablo Antolin Sanchez, Annalisa Buffa, Xiaodong Wei

We present a novel method to perform numerical integration over curved polyhedra enclosed by high-order parametric surfaces. Such a polyhedron is first decomposed into a set of triangular and/or rectangular pyramids, whose certain faces correspond to the given parametric surfaces. Each pyramid serves as an integration cell with a geometric mapping from a standard parent domain (e.g., a unit cube), where the tensor-product Gauss quadrature is adopted. As no constraint is imposed on the decomposition, certain resulting pyramids may intersect with themselves, and thus their geometric mappings may present negative Jacobian values. We call such cells folded cells and refer to the corresponding decomposition as a folded decomposition. We show that folded cells do not cause any issues in practice as they are only used to numerically compute certain integrals of interest. The same idea can be applied to planar curved polygons as well. We demonstrate both theoretically and numerically that folded cells can retain the same accuracy as the cells with strictly positive Jacobians. On the other hand, folded cells allow for a much easier and much more flexible decomposition for general curved polyhedra, on which one can robustly compute integrals. In the end, we show that folded cells can flexibly and robustly accommodate real-world complex geometries by presenting several examples in the context of immersed isogeometric analysis, where involved sharp features can be well respected in generating integration cells.(c) 2022 Elsevier B.V. All rights reserved.

ELSEVIER SCIENCE SA2022

Isogeometric analysis (IGA) was introduced to integrate methods for analysis and computer-aided design (CAD) into a unified process. High-quality parameterization of a physical domain plays a crucial role in IGA. However, obtaining high-quality parameterizations for complex geometries retains a challenge. Triangle configuration based bivariate simplex splines (TCB-splines) and their rational extensions provide an attractive alternative to classical nonuniform rational B-splines (NURBS) in the context of IGA, as they not only share many desired properties with NURBS but also can be defined over general polygonal domains. In this work, using TCB-splines in IGA is suggested to overcome the limits posed by the tensor-product structure of NURBS. We present the methodology for IGA to use rational TCB-splines over general polygonal domains. Then a method to generate the parametric domain according to given physical domain boundaries is proposed. This allows us to obtain a high-quality parameterization easily without resorting to the optimization method. Several numerical examples with complex physical domains demonstrate the flexibility of our TCB-spline-based IGA method, the high quality of the parameterization, and the accuracy of the numerical solutions. (C) 2022 Elsevier B.V. All rights reserved.

ELSEVIER SCIENCE SA2022

Geometry

Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest b

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathema

Finite element method

The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the tr