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 objec ...

. Before the emergence of

computational science Computational science, also known as scientific computing or scientific computation (SC), is a field in mathematics that uses advanced computing capabilities to understand and solve complex problems. It is an area of science that spans many disc ...

(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 Applied mechanics is the branch of science concerned with the motion of any substance that can be experienced or perceived by humans without the help of instruments. In short, when mechanics concepts surpass being theoretical and are applied and e ...

. 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 an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 Applied science, practical discipli ...

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 ...

computational thermodynamics Computational thermodynamics is the use of computers to simulate thermodynamic problems specific to materials science, particularly used in the construction of phase diagrams. Several open and commercial programs exist to perform these operation ...

computational electromagnetics Computational electromagnetics (CEM), computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic fields with physical objects and the environment. It typically involves using computer ...

, computational solid mechanics 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 imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...

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 matrice ...

and

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

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

finite difference A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for t ...

, 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 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

s. 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 ...

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 numerica ...

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. Computer programs are one component of software, which also includes software documentation, documentation and oth ...

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,

supercomputer A supercomputer is a 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 instructio ...

s or

parallel computers Parallel computing is a type of computation in which many calculations or processes are carried out simultaneously. Large problems can often be divided into smaller ones, which can then be solved at the same time. There are several different for ...

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 solution Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic or analytical can also have the following meanings: Chemistry * ...

s 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