Combining Polygon Schematization and Decomposition Approaches for Solving the Cavity Decomposition Problem

The cavity decomposition problem is a computational geometry problem, arising in the context of modern electronic CAD systems, that concerns detecting the generation and propagation of electromagnetic noise into multi-layer printed circuit boards. Algorithmically speaking, the problem can be formulated so as to contain, as sub-problems, the well-known polygon schematization and polygon decomposition problems. Given a polygon P and a finite set C of given directions, polygon schematization asks for computing a C -oriented polygon P ′ with “low complexity” and “high resemblance” to P , whereas polygon decomposition asks for partitioning P into a set of basic polygonal elements (e.g., triangles) whose size is as small as possible. In this article, we present three different solutions for the cavity decomposition problem, which are obtained by suitably combining existing algorithms for polygon schematization and decomposition, by considering different input parameters, and by addressing both methodological and implementation issues. Since it is difficult to compare the three solutions on a theoretical basis, we present an extensive experimental study, employing both real-world and random data, conducted to assess their performance. We rank the proposed solutions according to the results of the experimental evaluation, and provide insights on natural candidates to be adopted, in practice, as modules of modern printed circuit board design software tools, depending on the observed performance and on the different constraints on the desired output.

Enhanced numerical integration scheme based on image compression techniques: Application to rational polygonal interpolants

Polygonal finite elements offer an increased freedom in terms of mesh generation at the price of more complex, often rational, shape functions. Thus, the numerical integration of rational interpolants over polygonal domains is one of the challenges that needs to be solved. If, additionally, strong discontinuities are present in the integrand, e.g., when employing fictitious domain methods, special integration procedures must be developed. Therefore, we propose to extend the conventional quadtree-decomposition-based integration approach by image compression techniques. In this context, our focus is on unfitted polygonal elements using Wachspress shape functions. In order to assess the performance of the novel integration scheme, we investigate the integration error and the compression rate being related to the reduction in integration points. To this end, the area and the stiffness matrix of a single element are computed using different formulations of the shape functions, i.e., global and local, and partitioning schemes. Finally, the performance of the proposed integration scheme is evaluated by investigating two problems of linear elasticity.

Simultaneous Smoothing and Untangling of 2D Meshes Based on Explicit Element Geometric Transformation and Element Stitching

Mesh quality can affect both the accuracy and efficiency of numerical solutions. This paper first proposes a geometry-based smoothing and untangling method for 2D meshes based on explicit element geometric transformation and element stitching. A new explicit element geometric transformation (EEGT) operation for polygonal elements is firstly presented. The transformation, if applied iteratively to an arbitrary polygon (even inverted), will improve its regularity and quality. Then a well-designed element stitching scheme is introduced, which is achieved by carefully choosing appropriate element weights to average the temporary nodes obtained by the above individual element transformation. Based on the explicit element geometric transformation and element stitching, a new mesh smoothing and untangling approach for 2D meshes is proposed. The proper choice of averaging weights for element stitching ensures that the elements can be transitioned smoothly and uniformly throughout the calculation domain. Numerical results show that the proposed method is able to produce high-quality meshes with no inverted elements for highly tangled meshes. Besides, the inherent regularity and fine-grained parallelism make it suitable for implementation on Graphic Processor Unit (GPU).

A virtual element method for frictional contact including large deformations

Purpose This paper aims to describe the application of the virtual element method (VEM) to contact problems between elastic bodies. Design/methodology/approach Polygonal elements with arbitrary shape allow a stable node-to-node contact enforcement. By adaptively adjusting the polygonal mesh, this methodology is extended to problems undergoing large frictional sliding. Findings The virtual element is well suited for large deformation contact problems. The issue of element stability for this specific application is discussed, and the capability of the method is demonstrated by means of numerical examples. Originality/value This work is completely new as this is the first time, as per the authors’ knowledge, the VEM is applied to large deformation contact.

Virtual element methods for the obstacle problem

We study virtual element methods (VEMs) for solving the obstacle problem, which is a representative elliptic variational inequality of the first kind. VEMs can be regarded as a generalization of standard finite element methods with the addition of some suitable nonpolynomial functions, and the degrees of freedom are carefully chosen so that the stiffness matrix can be computed without actually computing the nonpolynomial functions. With this special design, VEMS can easily deal with complicated element geometries. In this paper we establish a priori error estimates of VEMs for the obstacle problem. We prove that the lowest-order ($k=1$) VEM achieves the optimal convergence order, and suboptimal order is obtained for the VEM with $k=2$. Two numerical examples are reported to show that VEM can work on very general polygonal elements, and the convergence orders in the $H^1$ norm agree well with the theoretical prediction.