Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics, numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.
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.
The Portable, Extensible Toolkit for Scientific Computation, 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.
Abaqus FEA is a software suite for finite element analysis and computer-aided engineering, originally released in 1978. The name and logo of this software are based on the abacus calculation tool. The Abaqus product suite consists of five core software products:
- Abaqus/CAE, or "Complete Abaqus Environment". It is a software application used for both the modeling and analysis of mechanical components and assemblies (pre-processing) and visualizing the finite element analysis result. A subset of Abaqus/CAE including only the post-processing module can be launched independently in the Abaqus/Viewer product.
- Abaqus/Standard, a general-purpose Finite-Element analyzer that employs implicit integration scheme (traditional).
- Abaqus/Explicit, a special-purpose Finite-Element analyzer that employs explicit integration scheme to solve highly nonlinear systems with many complex contacts under transient loads.
- Abaqus/CFD, a Computational Fluid Dynamics software application which provides advanced computational fluid dynamics capabilities with extensive support for preprocessing and postprocessing provided in Abaqus/CAE - discontinued in Abaqus 2017 and further releases.
- Abaqus/Electromagnetic, a Computational electromagnetics software application which solves advanced computational electromagnetic problems.
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.
Hermes2D is a C++/Python library of algorithms for rapid development of adaptive hp-FEM solvers. hp-FEM is a modern version of the finite element method (FEM) that is capable of extremely fast, exponential convergence.
hp-FEM is a general version of the finite element method (FEM), a numerical method for solving partial differential equations based on piecewise-polynomial approximations that employs elements of variable size (h) and polynomial degree (p). The origins of hp-FEM date back to the pioneering work of Barna A. Szabó and Ivo Babuška who discovered 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 makes the method very attractive compared to most other finite element methods, which only converge with an algebraic rate. The exponential convergence of hp-FEM was not only predicted theoretically, but also observed by numerous independent researchers.
Z88 is a software package for the finite element method (FEM) and topology optimization. A team led by Frank Rieg at the University of Bayreuth started development in 1985 and now the software is used by several universities, as well as small and medium-sized enterprises. Z88 is capable of calculating two and three dimensional element types with a linear approach. The software package contains several solvers and two post-processors and is available for Microsoft Windows, Mac OS X and Unix/Linux computers in 32-bit and 64-bit versions. Benchmark tests conducted in 2007 showed a performance on par with commercial software.
Klaus-Jürgen Bathe is a civil engineer, professor of mechanical engineering at the Massachusetts Institute of Technology, and founder of ADINA R&D, who specializes in computational mechanics. Bathe is considered to be one of the pioneers in the field of finite element analysis and its applications.
FEBio(Finite Elements for Biomechanics) is a software package for finite element analysis and was specifically designed for applications in biomechanics and bioengineering. It was developed in collaboration with research groups from the University of Utah and Columbia University (MBL).
Smoothed finite element methods (S-FEM) are a particular class of numerical simulation algorithms for the simulation of physical phenomena. It was developed by combining meshfree methods with the finite element method. S-FEM are applicable to solid mechanics as well as fluid dynamics problems, although so far they have mainly been applied to the former.
VisualFEA is a finite element analysis software program for Microsoft Windows and Mac OS X. It is developed and distributed by Intuition Software, Inc. of South Korea, and used chiefly for structural and geotechnical analysis. Its strongest point is its intuitive, user-friendly design based on graphical pre- and postprocessing capabilities. It has educational features for teaching and learning structural mechanics, and finite element analysis through graphical simulation. It is widely used in college-level courses related to structural mechanics and finite element methods.
SLEPc is a software library for the parallel computation of eigenvalues and eigenvectors of large, sparse matrices. It can be seen as a module of PETSc that provides solvers for different types of eigenproblems, including linear and nonlinear, as well as the SVD. Recent versions also include support for matrix functions. It uses the MPI standard for parallelization. Both real and complex arithmetic are supported, with single, double and quadruple precision.
GetFEM++ is a generic finite element C++ library with interfaces for Python, Matlab and Scilab. It aims at providing finite element methods and elementary matrix computations for solving linear and non-linear problems numerically. Its flexibility in choosing among different finite element approximations and numerical integration methods is one of its distinguishing characteristics.
MOOSE is an object-oriented C++ finite element framework for the development of tightly coupled multiphysics solvers from Idaho National Laboratory. MOOSE makes use of the PETSc non-linear solver package and libmesh to provide the finite element discretization.
Goma is an open-source, parallel, and scalable multiphysics software package for modeling and simulation of real-life physical processes, with a basis in computational fluid dynamics for problems with evolving geometry. It solves problems in all branches of mechanics, including fluids, solids, and thermal analysis. Goma uses advanced numerical methods, focusing on the low-speed flow regime with coupled phenomena for manufacturing and performance applications. It also provides a flexible software development environment for specialty physics.
