As the use of spectral/$hp$ element methods, and high-order finite element methods in general, continues to spread, community efforts to create efficient, optimized algorithms associated with fundamental high-order operations have grown. Core tasks such as solution expansion evaluation at quadrature points, stiffness and mass matrix generation, and matrix assembly have received tremendousattention. With the expansion of the types of problems to which high-order methods are applied, and correspondingly the growth in types of numerical tasks accomplished through high-order methods, the number and types of these core operations broaden. This work focuses on solution expansion evaluation at arbitrary points within an element. This operation is core to many postprocessing applications such as evaluation of streamlines and pathlines, as well as to field projection techniques such as mortaring. We expand barycentric interpolation techniques developed on an interval to 2D (triangles and quadrilaterals) and 3D (tetrahedra, prisms, pyramids, and hexahedra) spectral/$hp$ element methods. We provide efficient algorithms for their implementations, and demonstrate their effectiveness using the spectral/$hp$ element library Nektar++.
In this paper, applied strictly monotonic increasing scaled maps, a kind of well-conditioned linear barycentric rational interpolations are proposed to approximate functions of singularities at the origin, such as $x^α$ for $α\in (0,1)$ and $\log(x)$. It just takes $O(N)$ flops and can achieve fast convergence rates with the choice the scaled parameter, where $N$ is the maximum degree of the denominator and numerator. The construction of the rational interpolant couples rational polynomials in the barycentric form of second kind with the transformed Jacobi-Gauss-Lobatto points. Numerical exper…
Boundary element methods (BEM) reduce a partial differential equation in a domain to an integral equation on the domain's boundary. They are particularly attractive for solving problems on unbounded domains, but handling the dense matrices corresponding to the integral operators requires efficient algorithms. This article describes two approaches that allow us to solve boundary element equations on surface meshes consisting of several millions of triangles while preserving the optimal convergence rates of the Galerkin discretization.
On the finite element approximation for fractional fast diffusion equations
考虑$\mathbb{R}^N$中有界开多面体区域上的分数阶快速扩散方程，利用$C^0$ --空间上的分段线性有限元和时间上的向后Euler离散，给出了解的完全Galerkin逼近，证明了近似解的先验估计和收敛速度，推广了\emph{Carsten Ebmeyer and Wen Bin Liu, SIAM J. Numer. Anal., 46(2008), pp. 2393--2410}的结果.在\emph{Qiang Du, Max Gunzburger, Richaed B. Lehoucq and Kun Zhou, SIAM Rev., 54 (2012), no. 4, pp. 667--696.}的框架下，我们将先验估计和收敛速度推广到抛物型积分方程
The direct and indirect boundary element methods, accelerated via the fast multipole method, are applied to numerical simulation of room acoustics for large rooms of volume $\sim 150$ $m^{3}$ and frequencies up to 5 kHz on a workstation. As the parameter $kD$ (wavenumber times room diameter) is large, stabilization of the previously developed FMM algorithms is required for accuracy. A stabilization scheme is one of the key contribution of this paper. The computations are validated using well-known image source solutions for shoebox shaped rooms. Computations for L-shaped rooms are performed to…
The finite element method is a well-established method for the numerical solution of partial differential equations (PDEs), both linear and nonlinear. However, the repeated reassemblage of finite element matrices for nonlinear PDEs is frequently pointed out as one of the bottlenecks in the computations. The second bottleneck being the large and numerous linear systems to be solved arising from the use of Newton's method to solve nonlinear systems of equations. In this paper, we will address the first issue. We will see how under mild assumptions the assemblage procedure may be rewritten using …
Three algorithm are proposed to evaluate volume potentials that arise in boundary element methods for elliptic PDEs. The approach is to apply a modified fast multipole method for a boundary concentrated volume mesh. If $h$ is the meshwidth of the boundary, then the volume is discretized using nearly $O(h^{-2})$ degrees of freedom, and the algorithm computes potentials in nearly $O(h^{-2})$ complexity. Here nearly means that logarithmic terms of $h$ may appear. Thus the complexity of volume potentials calculations is of the same asymptotic order as boundary potentials. For sources and potential…
Embedded symmetric positive semi-definite machine-learned elements for reduced-order modeling in finite-element simulations with application to threaded fasteners
We present a novel approach to the parallelization of the parabolic fast multipole method for a space-time boundary element method for the heat equation. We exploit the special temporal structure of the involved operators to provide an efficient distributed parallelization with respect to time and with a one-directional communication pattern. On top, we apply a task-based shared memory parallelization and SIMD vectorization. In the numerical tests we observe high efficiencies of our parallelization approach.
We introduce a fast direct solver for variable-coefficient elliptic partial differential equations on surfaces based on the hierarchical Poincaré-Steklov method. The method takes as input an unstructured, high-order quadrilateral mesh of a surface and discretizes surface differential operators on each element using a high-order spectral collocation scheme. Elemental solution operators and Dirichlet-to-Neumann maps tangent to the surface are precomputed and merged in a pairwise fashion to yield a hierarchy of solution operators that may be applied in $\mathcal{O}(N \log N)$ operations for a mes…
In finite element calculations, the integral forms are usually evaluated using nested loops over elements, and over quadrature points. Many such forms (e.g. linear or multi-linear) can be expressed in a compact way, without the explicit loops, using a single tensor contraction expression by employing the Einstein summation convention. To automate this process and leverage existing high performance codes, we first introduce a notation allowing trivial differentiation of multi-linear finite element forms. Based on that we propose and describe a new transpiler from Einstein summation based expres…
In this work, two fast multipole boundary element formulations for the linear time-harmonic acoustic analysis of finite periodic structures are presented. Finite periodic structures consist of a bounded number of unit cell replications in one or more directions of periodicity. Such structures can be designed to efficiently control and manipulate sound waves and are referred to as acoustic metamaterials or sonic crystals. Our methods subdivide the geometry into boxes which correspond to the unit cell. A boundary element discretization is applied and interactions between well separated boxes are…
Our objective is to stabilise and accelerate the time-domain boundary element method (TDBEM) for the three-dimensional wave equation. To overcome the potential time instability, we considered using the Burton--Miller-type boundary integral equation (BMBIE) instead of the ordinary boundary integral equation (OBIE), which consists of the single- and double-layer potentials. In addition, we introduced a smooth temporal basis, i.e. the B-spline temporal basis of order $d$, whereas $d=1$ was used together with the OBIE in a previous study [Takahashi 2014]. Corresponding to these new techniques, we …
