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MFEM
MFEM is an open-source C++ library for solving partial differential equations using the finite element method, developed and maintained by researchers at the Lawrence Livermore National Laboratory and the MFEM open-source community on GitHub. MFEM is free software released under a BSD license. The library consists of C++ classes that serve as building blocks for developing finite element solvers applicable to problems of fluid dynamics, structural mechanics, electromagnetics, radiative transfer and many other. Features Some of the features of MFEM include * Arbitrary high order finite elements with curved boundaries. * H1, H(curl) and H(div) conforming, discontinuous (L2), and NURBS finite element spaces. * Local mesh refinement, both conforming (simplex meshes) and non-conforming ( quadrilateral/hexahedral meshes). * Highly scalable MPI-based parallelism and GPU acceleration. * Wide variety of finite element discretization approaches, including Galerkin, discontinuo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linux
Linux ( or ) is a family of open-source Unix-like operating systems based on the Linux kernel, an operating system kernel first released on September 17, 1991, by Linus Torvalds. Linux is typically packaged as a Linux distribution, which includes the kernel and supporting system software and libraries, many of which are provided by the GNU Project. Many Linux distributions use the word "Linux" in their name, but the Free Software Foundation uses the name "GNU/Linux" to emphasize the importance of GNU software, causing some controversy. Popular Linux distributions include Debian, Fedora Linux, and Ubuntu, the latter of which itself consists of many different distributions and modifications, including Lubuntu and Xubuntu. Commercial distributions include Red Hat Enterprise Linux and SUSE Linux Enterprise. Desktop Linux distributions include a windowing system such as X11 or Wayland, and a desktop environment such as GNOME or KDE Plasma. Distributions intended for ser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, * a 0-dimensional simplex is a point, * a 1-dimensional simplex is a line segment, * a 2-dimensional simplex is a triangle, * a 3-dimensional simplex is a tetrahedron, and * a 4-dimensional simplex is a 5-cell. Specifically, a ''k''-simplex is a ''k''-dimensional polytope which is the convex hull of its ''k'' + 1 vertices. More formally, suppose the ''k'' + 1 points u_0, \dots, u_k \in \mathbb^ are affinely independent, which means u_1 - u_0,\dots, u_k-u_0 are linearly independent. Then, the simplex determined by them is the set of points : C = \left\ This representation in terms of weighted vertices is known as the barycentric coordinate system. A regular simplex is a simplex that is also a regular polytope. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
UMFPACK
UMFPACK () is a set of routines for solving unsymmetric sparse linear systems of the form Ax=b, using the Unsymmetric MultiFrontal method (Matrix A is not required to be symmetric). Written in ANSI/ISO C and interfaces with * MATLAB version 6.0 and later * SciPy, and thus SciPy-relied software FuncDesigner, SageMath, PythonXY It appears as a built-in routine (for lu, backslash, and forward slash) in MATLAB, and includes a MATLAB interface, a C-callable interface, and a Fortran-callable interface. Note that "UMFPACK" is pronounced in two syllables, "Umph Pack". It is not "You Em Ef Pack" . UMFPACK has installation options to use the many versions of the BLAS, or no BLAS at all. BLAS is what UMFPACK relies on, to get high performance on a wide range of computers. Versions 1 and 1.1 were in Fortran 77 (Jan. 1995) and are licensed for non-commercial only. Version 2.2 appears as the Fortran package MA38 in the Harwell Subroutine Library. Versions 3 (March 2001) to 5.1 (May 2007 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PETSc
The Portable, Extensible Toolkit for Scientific Computation (PETSc, pronounced PET-see; the S is silent), is a suite of data structures and routines developed by Argonne National Laboratory for the scalable (parallel) solution of scientific applications modeled by partial differential equations. It employs the Message Passing Interface (MPI) standard for all message-passing communication. PETSc is the world’s most widely used parallel numerical software library for partial differential equations and sparse matrix computations. PETSc received an R&D 100 Award in 2009. The PETSc Core Development Group won the SIAM/ACM Prize in Computational Science and Engineering for 2015. PETSc is intended for use in large-scale application projects, many ongoing computational science projects are built around the PETSc libraries. Its careful design allows advanced users to have detailed control over the solution process. PETSc includes a large suite of parallel linear and nonlinear equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypre
The Parallel High Performance Preconditioners (hypre) is a library of routines for scalable (parallel) solution of linear systems. The built-in BLOPEX package in addition allows solving eigenvalue problems. The main strength of Hypre is availability of high performance parallel multigrid preconditioners for both structured and unstructured grid problems, see (Falgout et al., 2005, 2006). Currently, Hypre supports only real double-precision arithmetic. Hypre uses the Message Passing Interface (MPI) standard for all message-passing communication. PETSc has an interface to call Hypre preconditioners. Hypre is being developed and is supported by members of the Scalable Linear Solvers project within the Lawrence Livermore National Laboratory. Features hypre provides the following features: * Parallel vectors and matrices, using several different interfaces * Scalable parallel preconditioners * Built-in BLOPEX Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isogeometric Analysis
Isogeometric analysis is a computational approach that offers the possibility of integrating finite element analysis (FEA) into conventional NURBS-based CAD design tools. Currently, it is necessary to convert data between CAD and FEA packages to analyse new designs during development, a difficult task since the two computational geometric approaches are different. Isogeometric analysis employs complex NURBS geometry (the basis of most CAD packages) in the FEA application directly. This allows models to be designed, tested and adjusted in one go, using a common data set. The pioneers of this technique are Tom Hughes and his group at The University of Texas at Austin. A reference free software implementation of some isogeometric analysis methods is GeoPDEs. Likewise, other implementations can be found online. For instance, PetIGA is an open framework for high performance isogeometric analysis heavily based on PETSc. In addition, MIGFEM is another IGA code which is implemented in Mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Element Method
In the numerical solution of partial differential equations, a topic in mathematics, the spectral element method (SEM) is a formulation of the finite element method (FEM) that uses high degree piecewise polynomials as basis functions. The spectral element method was introduced in a 1984 paper by A. T. Patera. Although Patera is credited with development of the method, his work was a rediscovery of an existing method (see Development History) Discussion The spectral method expands the solution in trigonometric series, a chief advantage being that the resulting method is of a very high order. This approach relies on the fact that trigonometric polynomials are an orthonormal basis for L^2(\Omega). The spectral element method chooses instead a high degree piecewise polynomial basis functions, also achieving a very high order of accuracy. Such polynomials are usually orthogonal Chebyshev polynomials or very high order Lagrange polynomials over non-uniformly spaced nodes. In SEM co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixed Finite Element Method
In numerical analysis, the mixed finite element method, is a type of finite element method in which extra fields to be solved are introduced during the posing a partial differential equation problem. Somewhat related is the hybrid finite element method. The extra fields are constrained by using Lagrange multiplier fields. To be distinguished from the mixed finite element method, usual finite element methods that do not introduce such extra fields are also called irreducible or primal finite element methods. The mixed finite element method is efficient for some problems that would be numerically ill-posed if discretized by using the irreducible finite element method; one example of such problems is to compute the stress and strain fields in an almost incompressible elastic body. In mixed methods, the Lagrange multiplier fields inside the elements, usually enforcing the applicable partial differential equations. This results in a saddle point system having negative pivots and eigenval ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discontinuous Galerkin Method
In applied mathematics, discontinuous Galerkin methods (DG methods) form a class of numerical methods for solving differential equations. They combine features of the finite element and the finite volume framework and have been successfully applied to hyperbolic, elliptic, parabolic and mixed form problems arising from a wide range of applications. DG methods have in particular received considerable interest for problems with a dominant first-order part, e.g. in electrodynamics, fluid mechanics and plasma physics. Discontinuous Galerkin methods were first proposed and analyzed in the early 1970s as a technique to numerically solve partial differential equations. In 1973 Reed and Hill introduced a DG method to solve the hyperbolic neutron transport equation. The origin of the DG method for elliptic problems cannot be traced back to a single publication as features such as jump penalization in the modern sense were developed gradually. However, among the early influential contribut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galerkin Method
In mathematics, in the area of numerical analysis, Galerkin methods, named after the Russian mathematician Boris Galerkin, convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete problem by applying linear constraints determined by finite sets of basis functions. Often when referring to a Galerkin method, one also gives the name along with typical assumptions and approximation methods used: * Ritz–Galerkin method (after Walther Ritz) typically assumes symmetric and positive definite bilinear form in the weak formulation, where the differential equation for a physical system can be formulated via minimization of a quadratic function representing the system energy and the approximate solution is a linear combination of the given set of the basis functions.A. Ern, J.L. Guermond, ''Theory and practice of finite elements'', Springer, 2004, * Bubnov–Galerkin method (after Ivan Bubnov) does not require the bilinear fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
YouTube
YouTube is a global online video platform, online video sharing and social media, social media platform headquartered in San Bruno, California. It was launched on February 14, 2005, by Steve Chen, Chad Hurley, and Jawed Karim. It is owned by Google, and is the List of most visited websites, second most visited website, after Google Search. YouTube has more than 2.5 billion monthly users who collectively watch more than one billion hours of videos each day. , videos were being uploaded at a rate of more than 500 hours of content per minute. In October 2006, YouTube was bought by Google for $1.65 billion. Google's ownership of YouTube expanded the site's business model, expanding from generating revenue from advertisements alone, to offering paid content such as movies and exclusive content produced by YouTube. It also offers YouTube Premium, a paid subscription option for watching content without ads. YouTube also approved creators to participate in Google's Google AdSens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gpu Computing
General-purpose computing on graphics processing units (GPGPU, or less often GPGP) is the use of a graphics processing unit (GPU), which typically handles computation only for computer graphics, to perform computation in applications traditionally handled by the central processing unit (CPU). The use of multiple video cards in one computer, or large numbers of graphics chips, further parallelizes the already parallel nature of graphics processing. Essentially, a GPGPU pipeline is a kind of parallel processing between one or more GPUs and CPUs that analyzes data as if it were in image or other graphic form. While GPUs operate at lower frequencies, they typically have many times the number of cores. Thus, GPUs can process far more pictures and graphical data per second than a traditional CPU. Migrating data into graphical form and then using the GPU to scan and analyze it can create a large speedup. GPGPU pipelines were developed at the beginning of the 21st century for graphic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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