Numerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs).
The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations, including fluid mechanics, acoustics, electromagnetics, fracture mechanics, and contact mechanics.
COMSOL Multiphysics is a finite element analysis, solver, and simulation software package for various physics and engineering applications, especially coupled phenomena and multiphysics. The software facilitates conventional physics-based user interfaces and coupled systems of partial differential equations (PDEs). COMSOL provides an IDE and unified workflow for electrical, mechanical, fluid, acoustics, and chemical applications.
Mesh generation is the practice of creating a mesh, a subdivision of a continuous geometric space into discrete geometric and topological cells. Often these cells form a simplicial complex. Usually the cells partition the geometric input domain. Mesh cells are used as discrete local approximations of the larger domain. Meshes are created by computer algorithms, often with human guidance through a GUI, depending on the complexity of the domain and the type of mesh desired. A typical goal is to create a mesh that accurately captures the input domain geometry, with high-quality (well-shaped) cells, and without so many cells as to make subsequent calculations intractable. The mesh should also be fine in areas that are important for the subsequent calculations.
Computational electromagnetics (CEM), computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic fields with physical objects and the environment using computers.
Dassault Systèmes Simulia Corp. is a computer-aided engineering (CAE) vendor. Formerly known as Abaqus Inc. and previously Hibbitt, Karlsson & Sorensen, Inc., (HKS), the company was founded in 1978 by David Hibbitt, Bengt Karlsson and Paul Sorensen, and has its headquarters in Providence, Rhode Island.
E The finite-difference frequency-domain (FDFD) method is a numerical solution method for problems usually in electromagnetism and sometimes in acoustics, based on finite-difference approximations of the derivative operators in the differential equation being solved.
Computational magnetohydrodynamics (CMHD) is a rapidly developing branch of magnetohydrodynamics that uses numerical methods and algorithms to solve and analyze problems that involve electrically conducting fluids. Most of the methods used in CMHD are borrowed from the well established techniques employed in Computational fluid dynamics. The complexity mainly arises due to the presence of a magnetic field and its coupling with the fluid. One of the important issues is to numerically maintain the (conservation of magnetic flux) condition, from Maxwell's equations, to avoid the presence of unrealistic effects, namely magnetic monopoles, in the solutions.
The FEniCS Project is a collection of free and open-source software components with the common goal to enable automated solution of differential equations. The components provide scientific computing tools for working with computational meshes, finite-element variational formulations of ordinary and partial differential equations, and numerical linear algebra.
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.
hp-FEM is a generalization of the finite element method (FEM) for solving partial differential equations numerically based on piecewise-polynomial approximations employing elements of variable size (h) and polynomial degree (p). hp-FEM originates from the discovery by Barna A. Szabó and Ivo Babuška that the finite element method converges exponentially fast when the mesh is refined using a suitable combination of h-refinements (dividing elements into smaller ones) and p-refinements. The exponential convergence of hp-FEM has been observed by numerous independent researchers. On many problems, hp-FEM may be faster than finite element methods that converge at an algebraic rate.
Synopsys Simpleware ScanIP is a 3D image processing and model generation software program developed by Synopsys Inc. to visualise, analyse, quantify, segment and export 3D image data from magnetic resonance imaging (MRI), computed tomography (CT), microtomography and other modalities for computer-aided design (CAD), finite element analysis (FEA), computational fluid dynamics (CFD), and 3D printing. The software is used in the life sciences, materials science, nondestructive testing, reverse engineering and petrophysics.
Gmsh is a finite-element mesh generator developed by Christophe Geuzaine and Jean-François Remacle. Released under the GNU General Public License, Gmsh is free software.
In computational fluid dynamics, TELEMAC is short for the open TELEMAC-MASCARET system, or a suite of finite element computer program owned by the Laboratoire National d'Hydraulique et Environnement (LNHE), part of the R&D group of Électricité de France. After many years of commercial distribution, a Consortium was officially created in January 2010 to organize the open source distribution of the open TELEMAC-MASCARET system now available under GPLv3.
deal.II is a free, open-source library to solve partial differential equations using the finite element method. The current release is version 9.5, released in July 2023. It is one of the most widely used finite element libraries and provides comprehensive support for all aspects of the solution of partial differential equations. The founding authors of the project — Wolfgang Bangerth, Ralf Hartmann, and Guido Kanschat — won the 2007 J. H. Wilkinson Prize for Numerical Software for deal.II. However, it is a worldwide project with around a dozen "Principal Developers", but over the years several hundred people have contributed substantial pieces of code or documentation to the project.
Agros2D is an open-source code for numerical solutions of 2D coupled problems in technical disciplines. Its principal part is a user interface serving for complete preprocessing and postprocessing of the tasks. The processor is based on the library Hermes containing the most advanced numerical algorithms for monolithic and fully adaptive solution of systems of generally nonlinear and nonstationary partial differential equations (PDEs) based on hp-FEM. Both parts of the code are written in C++.
FEATool Multiphysics is a physics, finite element analysis (FEA), and partial differential equation (PDE) simulation toolbox. FEATool Multiphysics features the ability to model fully coupled heat transfer, fluid dynamics, chemical engineering, structural mechanics, fluid-structure interaction (FSI), electromagnetics, as well as user-defined and custom PDE problems in 1D, 2D (axisymmetry), or 3D, all within a graphical user interface (GUI) or optionally as script files. FEATool has been employed and used in academic research, teaching, and industrial engineering simulation contexts.
