Computational mechanics is the discipline concerned with the use of computational methods to study phenomena governed by the principles of

mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects ...

. Before the emergence of computational science (also called scientific computing) as a "third way" besides theoretical and experimental sciences, computational mechanics was widely considered to be a sub-discipline of applied mechanics. It is now considered to be a sub-discipline within computational science.

Overview

Computational mechanics (CM) is interdisciplinary. Its three pillars are

, mathematics, and

computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...

and physics.

Mechanics

Computational fluid dynamics Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate t ...

, computational thermodynamics, computational electromagnetics, computational solid mechanics are some of the many specializations within CM.

Mathematics

The areas of mathematics most related to computational mechanics are partial differential equations,

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matric ...

and numerical analysis. The most popular numerical methods used are the

finite element The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical models, mathematical modeling. Typical problem areas of interest include the traditional fields of struct ...

, finite difference, and

boundary element The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in ''boundary integral'' form), including fluid mechanics, acoustics, el ...

methods in order of dominance. In solid mechanics finite element methods are far more prevalent than finite difference methods, whereas in fluid mechanics, thermodynamics, and electromagnetism, finite difference methods are almost equally applicable. The boundary element technique is in general less popular, but has a niche in certain areas including acoustics engineering, for example.

Computer Science

With regard to computing, computer programming, algorithms, and parallel computing play a major role in CM. The most widely used programming language in the scientific community, including computational mechanics, is Fortran. Recently, C++ has increased in popularity. The scientific computing community has been slow in adopting C++ as the lingua franca. Because of its very natural way of expressing mathematical computations, and its built-in visualization capacities, the proprietary language/environment MATLAB is also widely used, especially for rapid application development and model verification.

Process

Scientists within the field of computational mechanics follow a list of tasks to analyze their target mechanical process: # A

mathematical model A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...

of the physical phenomenon is made. This usually involves expressing the natural or engineering system in terms of partial differential equations. This step uses

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...

to formalize a complex system. # The mathematical equations are converted into forms which are suitable for digital computation. This step is called discretization because it involves creating an approximate discrete model from the original continuous model. In particular, it typically translates a partial differential equation (or a system thereof) into a system of algebraic equations. The processes involved in this step are studied in the field of numerical analysis. #

Computer program A computer program is a sequence or set of instructions in a programming language for a computer to execute. Computer programs are one component of software, which also includes documentation and other intangible components. A computer progra ...

s are made to solve the discretized equations using direct methods (which are single step methods resulting in the solution) or

iterative method In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the pre ...

s (which start with a trial solution and arrive at the actual solution by successive refinement). Depending on the nature of the problem, supercomputers or parallel computers may be used at this stage. # The mathematical model, numerical procedures, and the computer codes are verified using either experimental results or simplified models for which exact analytical solutions are available. Quite frequently, new numerical or computational techniques are verified by comparing their result with those of existing well-established numerical methods. In many cases, benchmark problems are also available. The numerical results also have to be visualized and often physical interpretations will be given to the results.

Applications

Some examples where computational mechanics have been put to practical use are vehicle crash simulation, petroleum reservoir modeling, biomechanics, glass manufacturing, and semiconductor modeling. Complex systems that would be very difficult or impossible to treat using analytical methods have been successfully simulated using the tools provided by computational mechanics.

References

External links

United States Association for Computational Mechanics

{{DEFAULTSORT:Computational Mechanics Computational science Mechanics Computational fields of study Computational physics