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

mechanics Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ...

. Before the emergence of

computational science Computational science, also known as scientific computing, technical computing or scientific computation (SC), is a division of science, and more specifically the Computer Sciences, which uses advanced computing capabilities to understand and s ...

(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 Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, and

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

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 dynamics, fluid flows. Computers are used to perform the calculations required ...

, computational thermodynamics, computational electromagnetics, computational

solid mechanics Solid mechanics (also known as mechanics of solids) is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation (mechanics), deformation under the action of forces, temperature chang ...

are some of the many specializations within CM.

Mathematics

The areas of mathematics most related to computational mechanics are

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

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 matrix (mathemat ...

and

numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...

. The most popular numerical methods used are the finite element,

finite difference A finite difference is a mathematical expression of the form . Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation. The difference operator, commonly d ...

, and boundary element 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 MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...

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 an abstract and concrete, abstract description of a concrete system using mathematics, mathematical concepts and language of mathematics, language. The process of developing a mathematical model is termed ''mathematical m ...

of the physical phenomenon is made. This usually involves expressing the natural or engineering system in terms of

s. This step uses

physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...

to formalize a complex system. # The mathematical equations are converted into forms which are suitable for digital computation. This step is called

discretization In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numeri ...

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

. #

Computer program A computer program is a sequence or set of instructions in a programming language for a computer to Execution (computing), execute. It is one component of software, which also includes software documentation, documentation and other intangibl ...

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 Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''i''-th approximation (called an " ...

s (which start with a trial solution and arrive at the actual solution by successive refinement). Depending on the nature of the problem,

supercomputer A supercomputer is a type of computer with a high level of performance as compared to a general-purpose computer. The performance of a supercomputer is commonly measured in floating-point operations per second (FLOPS) instead of million instruc ...

s 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