# FINITE ELEMENT METHOD IN ENGINEERING CALCULATIONS. SAE-SYSTEMS IN MECHANICS

FEM is an acronym that stands for "finite element method". Hence, the term "FEM analysis" follows, i.e. analysis and solution of certain engineering problems using this method.

The following analogues of these abbreviations are widespread in English-language literature:

1. FEM (Finite Element Method).
2. FEA (Finite Element Analysis).

Nowadays, it is probably hard to find a technical field which does not make use of FEM. Here are just a few of the problems engineers solve with FEM:

mechanics of deformable solids (structural analysis)

Heat transfer

Fluid dynamics (fluid flow)

mass transfer

electrodynamics and others.

FEM computational model | Dystlab Store

Why FEM is more about mathematics than engineering

It is no secret that the world around us (in general) and engineering objects (in particular) are described by partial differential equations. As a rule, it is quite difficult to get an exact analytical solution of such equations, so calculators are forced to resort to numerical (approximate) methods of calculation. One of such tools is the finite element method.

The essence of the finite element method

A typical engineering problem in FEM starts with the preparation of a model which is a virtual analog of a real building structure, technological product, machine part, etc.

From the geometric point of view, the computational model is a field of points connected to each other by primitives (straight line segments, triangles, rectangles, etc.). Thus, a kind of a grid structure is formed - the geometry of the original structure is approximated by the grid superimposed on it and further work is performed not with the original system but with the resulting grid.

In addition to geometry, the primitives connecting the node points of the model also have known mechanical properties. This means that by linking the stiffnesses of all the mesh elements together (within the assumptions accepted in the model), it is possible to establish the stress-strain state of the entire system. Thus, the calculator can get any factors of interest - longitudinal and transverse forces, bending and torsional moments, stresses, deformations, etc.

The number of nodes and elements that make up the computational model is known in advance. For some complex systems it can be measured in thousands or even millions, but it is finite anyway. This circumstance, as well as the fact that the principle of "working" of each individual element of the system is known in advance, gave rise to the name - the finite element method. And the mesh itself is called, as a rule, the finite-element mesh.

FEM calculation model | Dystlab Store

In terms of mathematics

The modern interpretation of FEM is quite complex and even a cursory description thereof is beyond the scope of this publication. A few key points can, however, be pointed out.

From a mathematical point of view, the domain in which a solution of the FEM system of differential equations is sought is divided into subregions (elements) and, for each element, an approximating function of arbitrary form is selected. The simplest and "crudest" case is a polynomial of the first degree: outside the element the selected function is equal to zero, and on the boundaries (in the nodes) the function takes values that are the solution of the problem. Of course, they are unknown in advance. Coefficients of the approximating function polynomial are found from the conditions of equality of values of neighboring functions in nodes.

Then we make a system of linear algebraic equations in which the number of unknowns equals the number of degrees of freedom in the system (in the general case, it is six times the number of nodes in the grid). The size of the grid is limited not only by the specific problem, but also by the physical capabilities of the computer (first of all, by the size of available memory).

In scientific and technical literature, the theory of the finite element method is stated through the matrix calculus. The steps described above necessarily include the collection of stiffness matrices and masses of the structure. The stiffness matrix is a table of nodal reactions of a finite element to a singular perturbation of each of its nodes. Simply put, the stiffness matrix of a finite element is a system of relationships between all its points at the "mechanical" level. Knowing the local stiffness matrix for each individual element, the computer program (CAE) generates a global stiffness matrix by adding the stiffnesses of all the elements connected in the common (adjacent) nodes, taking into account their orientation in space. The result is a general system of relationships between all the nodes of the computational model.

The result is a general system of relationships between all the nodes of the model.

### Finite element programs

CAE is an English abbreviation that literally means Computer-Aided Engineering. It is a common name for software that solves various engineering tasks (analysis, calculations, simulations of various physical processes, etc.).

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In the overwhelming majority, the calculation modules of modern CAE-software are based on the numerical methods of differential equations solution, namely FEM.

CAE-systems are almost inextricably linked to CAD-systems (Computer-Aided Design), since strength assessment or dynamic analysis are part of a more general design cycle, product design. Often they are integrated into each other at the user interface level. The revolutionary significance of CAE for modern engineering and engineering is that CAE software verifies the performance of a product or design without significant time and expense for in-situ testing, because the finite element method-based program evaluates the behavior of a computer model under near real-world conditions.

Key areas of CAE:

FEA (Finite Element Analysis) - strain and stress analysis of parts and assemblies

CFD (Computer Fluid Dynamics) - computational fluid dynamics, thermal and fluid flow analysis

MBD (MultiBody Dynamics and Kinematics) - multimass solid dynamics, kinematics

Structural Optimization

Tools for analysis and simulation of casting, forming, punching and other technological processes

Traditionally, different industries use different software solutions (the finite element method itself, of course, does not change). Moreover: competing companies from the same field try to use, as a rule, different software solutions. The choice of one or another tool depends on a large number of factors (for example, range of tasks to be solved, interface, number of users, license cost, support, and many others). At the end of this article we will give some comments on this subject.

Let us randomly list the most popular CAE-software used in mechanical engineering.

FEM calculation model | Dystlab Store

ABAQUS

ABAQUS software is considered one of the leading software in the large industry. Offers exceptionally detailed documentation and a wide range of solution options. It allows you to write customized subroutines for almost all basic functions. The interface looks somewhat "old-school". For most tasks in ABAQUS, the language used is FORTRAN.

ANSYS

One of the oldest and most widely used software in the industry. Works well with nearly every kind of simulation found in the industry. The documentation covers probably up to 90% of possible problems. The ANSYS interface is more intuitive than in ABAQUS.

NASTRAN

NASA STRucture ANalysis. Developed by NASA and widely used in the aerospace industry. Nastran contains publicly open source code, so the solver is used in many other CAE packages. It is oriented first of all on structural analysis problems.

Siemens NX

Is a 3-in-1 system, i.e. CAD + CAE + CAM. Offers a modern interface and advanced features, including:

Support for the complete product cycle, starting with CAD (Modeling), moving on to CAE (Advanced Simulation) and ending with CAM;

An interface that is intuitive to engineers with CAD experience;

Synchronous Modeling tools that allow making changes to an existing or imported model as quickly and easily as possible;

the possibility of creating an "idealized" model, as well as a number of tools to prepare the model for calculation without affecting the underlying geometry;

extensive postprocessor capabilities at the stage of analysis and interpretation of simulation results, etc.

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