Adaptive refinement for unstructured T-splines with linear complexity

The BEM-based Finite Element Method is one of the promising strategies which are applicable for the approximation of boundary value problems on general polygonal and polyhedral meshes. The flexibility with respect to meshes arises from the implicit definition of trial functions in a Trefftz-like manner. These functions are treated locally by means of Boundary Element Methods (BEM). The following presentation deals with the formulation of higher order trial functions and their application in uniform and adaptive mesh refinement strategies. For the adaptive refinement, a residual based error estimate is used on general polygonal meshes for the higher order, conforming trial functions. The first numerical results, in the context of adaptive refinement, show optimal rates of convergence with respect to the number of degrees of freedom.

Chapter 10 - Subdivision Surfaces

SUMMARY We present a new numerical method to solve the heterogeneous anelastic seismic wave equations with arbitrary high order of accuracy in space and time on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. Using the velocity–stress formulation provides a linear hyperbolic system of equations with source terms that is completed by additional equations for the anelastic functions including the strain history of the material. These additional equations result from the rheological model of the generalized Maxwell body and permit the incorporation of realistic attenuation properties of viscoelastic material accounting for the behaviour of elastic solids and viscous fluids. The proposed method relies on the finite volume (FV) approach where cell-averaged quantities are evolved in time by computing numerical fluxes at the element interfaces. The basic ingredient of the numerical flux function is the solution of Generalized Riemann Problems at the element interfaces according to the arbitrary high-order derivatives (ADER) approach of Toro et al., where the initial data is piecewise polynomial instead of piecewise constant as it was in the original first-order FV scheme developed by Godunov. The ADER approach automatically produces a scheme of uniformly high order of accuracy in space and time. The high-order polynomials in space, needed as input for the numerical flux function, are obtained using a reconstruction operator acting on the cell averages. This reconstruction operator uses some techniques originally developed in the Discontinuous Galerkin (DG) Finite Element framework, namely hierarchical orthogonal basis functions in a reference element. In particular, in this article we pay special attention to underline the differences as well as the points in common with the ADER-DG schemes previously developed by the authors, especially concerning the MPI parallelization of both methods. The numerical convergence analysis demonstrates that the proposed FV schemes provide very high order of accuracy even on unstructured tetrahedral meshes while computational cost for a desired accuracy can be reduced when applying higher order reconstructions. Applications to a series of well-acknowledged elastic and anelastic test cases and comparisons with analytic and numerical reference solutions, obtained by different well-established numerical methods, confirm the performance of the proposed method. Therefore, the development of the highly accurate ADER-FV approach for tetrahedral meshes including viscoelastic material provides a novel, flexible and efficient numerical technique to approach 3-D wave propagation problems including realistic attenuation and complex geometry.

New a posteriori error estimator and adaptive mesh refinement strategy for 2-D crack problems

For the evaluation of gas entrainment phenomena in the sodium-cooled fast reactor (JSFR), a high-precision simulation method for gas-liquid two-phase flows on unstructured meshes have been developed. In this study, our high-precision volume-of-fluid algorithm on unstructured meshes is introduced into the two-dimensional unstructured adaptive mesh algorithm (described in the 1st report). For that purpose, an appropriate redistribution method for volume fraction values is established. In addition, a calculation procedure is formulated, in which momentum values at both gas and liquid phases are conserved respectively when velocity values at refined or merged cells are calculated. This new unstructured adaptive mesh algorithm for gas-liquid two-phase flows is verified by solving well-known slotted disk rotation problems. As a result, the present adaptive mesh algorithm succeeds in maintaining slotted disk shape after one rotation by relatively small number of cells. In addition, the dam break problem is also simulated. The simulation results showed that the gas-liquid interface is simulated with high resolution and the transient interfacial position is more accurate than the simulation result on the non-adaptive mesh.

In this paper, local polynomial property, global linear independence, and local linear dependence of the convolution of a B-spline and a refinable distribution supported on a Cantor-like set are studied.

Local linear independence

This paper discusses compressive sensing with sparse measurement matrices. Sparse matrices have several attractive properties, like low computational complexity in both encoding and recovery, easy incremental updates to signals, and low storage requirement, etc. Typical examples of existing algorithms for sparse signal recovery with sparse measurement matrices include convex relaxation, matching pursuit, and Bayesian framework based approaches. In this paper, we propose an alternative technique to this problem. The proposed technique has a linear recovery complexity and relatively good empirical behavior. In this technique, we employ a permutation-based multi-dimensional measurement matrix, which is composed of several sub-matrices, each consisting of a block-diagonal matrix and a random permutation matrix. The measurement symbols generated from the same permutation matrix are referred to as a dimension. Such a measurement matrix brings some useful features to the measurement symbols. Fully exploiting these features enables us to design a simple and effective recovery algorithm. The proposed recovery algorithm employs an iterative process. In each iteration, the algorithm looks for measurement symbols with certain features, and uses such features to recover the source symbols related to these measurements. An iterative process is then applied, along with an interference cancellation operation, to reconstruct all source symbols gradually. The complexity of the proposed algorithm is relatively low, which grows linearly with the source signal length. Numerical results show that the proposed technique empirically offers a much lower sketch length than ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -minimization-based convex relaxation and Bayesian framework based algorithms. It achieves the empirical lower bound of sketch length and linear recovery complexity at the same time.

In this paper, we present a new time-domain method that has a naturally diagonal mass matrix and thereby a strict linear computational complexity per time step, regardless of whether the discretization is a structured grid or an unstructured mesh. This property is obtained independent of the element shape used for discretization. No interpolations, projections, and mass lumping are required. The accuracy and stability of the proposed method are both theoretically guaranteed. In addition, no dual mesh is needed and the tangential continuity of the fields is satisfied across the element interface. The flexible framework of the proposed method also allows for a straightforward extension to higher order accuracy in both electric and magnetic fields. Numerical experiments have been conducted on a variety of unstructured triangular-element meshes. Correlations with analytical solutions and the time-domain finite-element method have validated the accuracy and generality of the proposed new time-domain method.

Mesh refinement for anisotropic diffusion in magnetized plasmas

In this article, a novel adaptive mesh refinement scheme based on trimmed hexahedral (TH) meshes is proposed to simulate phase‐field fracture in brittle materials. A regular hexahedral background mesh is adaptively refined using a balanced octree algorithm to resolve the length scale in phase‐field fracture models. Adaptively refined TH meshes are created by cutting octree background meshes with the boundary of an analysis domain. A multithreshold criterion for mesh refinement is proposed to accurately capture the evolution of the damage phase field. Shape functions for transition TH elements are developed to satisfy the compatibility across the interelement boundaries between TH elements with different refinement levels. Numerical results show that the present scheme is very efficient and effective to trace arbitrary evolving phase‐field cracks in three dimensions.

Highly detailed geometric models, which are represented as dense triangular meshes are becoming popular in computer graphics. Since such 3D meshes often have huge information, we require some methods to treat them efficiently in the 3D mesh processing such as, surface simplification, subdivision surface, curved surface approximation and morphing. In these applications, we often extract features of 3D meshes such as feature vertices and feature edges in preprocessing step. An automatic extraction method of feature edges is treated in this study. In order to realize the feature edge extraction method, we first introduce the concavity and convexity evaluation value. Then the histogram of the concavity and convexity evaluation value is used to separate the feature edge region. We apply a thinning algorithm, which is used in 2D binary image processing. It is shown that the proposed method can extract appropriate feature edges from 3D meshes.

A finite-element ocean model: principles and evaluation

A method for generating moving, orthogonal, area preserving polygonal meshes

An ${\cal H}^{2}$ -matrix based linear complexity direct matrix solution is developed for the volume integral equation (VIE) based broadband full-wave extraction of general 3-D circuits. Such circuits are in general electrically small or moderate, but contain arbitrarily shaped lossy conductors immersed in inhomogeneous dielectrics with ports located anywhere in the physical layout of the circuit. In the proposed direct solver, we first develop a well-conditioned VIE formulation without sacrificing the rigor and the advantages of the prevailing formulations. This formulation facilitates a robust direct solution of good accuracy even with a rank-1 representation. We then overcome the numerical challenge of solving the resultant highly unstructured system matrix mixed with both square and rectangular dense and sparse matrices by developing a fast linear complexity direct solution. This direct solution is capable of inverting dense matrices involving over 2 million unknowns in less than 1 h on a single CPU core running at 3 GHz. Numerical simulations of large-scale 3-D circuits and comparisons with state-of-the-art linear complexity iterative VIE solvers have demonstrated the accuracy, efficiency, and linear complexity of the proposed direct VIE solver.