1,710 research outputs found
Computing hypergeometric functions rigorously
We present an efficient implementation of hypergeometric functions in
arbitrary-precision interval arithmetic. The functions , ,
and (or the Kummer -function) are supported for
unrestricted complex parameters and argument, and by extension, we cover
exponential and trigonometric integrals, error functions, Fresnel integrals,
incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre
functions, Jacobi polynomials, complete elliptic integrals, and other special
functions. The output can be used directly for interval computations or to
generate provably correct floating-point approximations in any format.
Performance is competitive with earlier arbitrary-precision software, and
sometimes orders of magnitude faster. We also partially cover the generalized
hypergeometric function and computation of high-order parameter
derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case
E-G in section 8.5 (table 6, figure 2); adjusted paper siz
Identifying combinations of tetrahedra into hexahedra: a vertex based strategy
Indirect hex-dominant meshing methods rely on the detection of adjacent
tetrahedra an algorithm that performs this identification and builds the set of
all possible combinations of tetrahedral elements of an input mesh T into
hexahedra, prisms, or pyramids. All identified cells are valid for engineering
analysis. First, all combinations of eight/six/five vertices whose connectivity
in T matches the connectivity of a hexahedron/prism/pyramid are computed. The
subset of tetrahedra of T triangulating each potential cell is then determined.
Quality checks allow to early discard poor quality cells and to dramatically
improve the efficiency of the method. Each potential hexahedron/prism/pyramid
is computed only once. Around 3 millions potential hexahedra are computed in 10
seconds on a laptop. We finally demonstrate that the set of potential hexes
built by our algorithm is significantly larger than those built using
predefined patterns of subdivision of a hexahedron in tetrahedral elements.Comment: Preprint submitted to CAD (26th IMR special issue
Supercomputer implementation of finite element algorithms for high speed compressible flows
Prediction of compressible flow phenomena using the finite element method is of recent origin and considerable interest. Two shock capturing finite element formulations for high speed compressible flows are described. A Taylor-Galerkin formulation uses a Taylor series expansion in time coupled with a Galerkin weighted residual statement. The Taylor-Galerkin algorithms use explicit artificial dissipation, and the performance of three dissipation models are compared. A Petrov-Galerkin algorithm has as its basis the concepts of streamline upwinding. Vectorization strategies are developed to implement the finite element formulations on the NASA Langley VPS-32. The vectorization scheme results in finite element programs that use vectors of length of the order of the number of nodes or elements. The use of the vectorization procedure speeds up processing rates by over two orders of magnitude. The Taylor-Galerkin and Petrov-Galerkin algorithms are evaluated for 2D inviscid flows on criteria such as solution accuracy, shock resolution, computational speed and storage requirements. The convergence rates for both algorithms are enhanced by local time-stepping schemes. Extension of the vectorization procedure for predicting 2D viscous and 3D inviscid flows are demonstrated. Conclusions are drawn regarding the applicability of the finite element procedures for realistic problems that require hundreds of thousands of nodes
Arbitrary order 2D virtual elements for polygonal meshes: Part II, inelastic problem
The present paper is the second part of a twofold work, whose first part is
reported in [3], concerning a newly developed Virtual Element Method (VEM) for
2D continuum problems. The first part of the work proposed a study for linear
elastic problem. The aim of this part is to explore the features of the VEM
formulation when material nonlinearity is considered, showing that the accuracy
and easiness of implementation discovered in the analysis inherent to the first
part of the work are still retained. Three different nonlinear constitutive
laws are considered in the VEM formulation. In particular, the generalized
viscoplastic model, the classical Mises plasticity with isotropic/kinematic
hardening and a shape memory alloy (SMA) constitutive law are implemented. The
versatility with respect to all the considered nonlinear material constitutive
laws is demonstrated through several numerical examples, also remarking that
the proposed 2D VEM formulation can be straightforwardly implemented as in a
standard nonlinear structural finite element method (FEM) framework
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