##### (A to Z) of Finite Element Analysis

#### A

**ACCELERATION**

The second time derivative of the displacement (the first time derivative of the velocity).

**ADAPTIVE FINITE ELEMENT METHOD/ADAPTIVE MESHING**

An adaptive finite element solver iteratively performs finite element analysis, determines the areas of the mesh where the solution is not sufficiently accurate and refines the mesh in those areas until the solution obtains the prescribed degree of accuracy. Adaptive Meshing involves automatically improving the mesh where necessary to meet specified convergence criteria.

**ALGEBRAIC EIGENVALUE PROBLEM**

The eigenvalue problem when written in the form of stiffness times mode shape minus eigenvalue times mass times mode shape is equal to zero. It is the form that arises naturally from a discrete parameter model in free vibration.

**ASPECT RATIO**

The ratio of the longest to shortest side lengths on an element.

**ASSEMBLY**

**Geometric:**Two or more parts mated together.

**FEA:** The process of assembling the element matrices together to form the global matrix.

Typically element stiffness matrices are assembled to form the complete stiffness matrix of the structure.

**AUTOMATIC MESH GENERATION**

The process of generating a mesh of elements over the volume that is being analyzed. There are two forms of automatic mesh generation: Free Meshing -Where the mesh has no structure to it. Free meshing generally uses triangular and tetrahedral elements.

Mapped Meshing -Where large regions, if not all, of the volume is covered with regular meshes. This can use any form of element. Free meshing can be used to fill any shape. Mapped meshing can only be used on some shapes without elements being excessively distorted.

**AXISYMMETRY**

If a shape can be defined by rotating a cross- section about a line (e.g. a cone) then it is said to be axisymmetric. This can be used to simplify the analysis of the system. Such models are sometimes called two and a half dimensional since a 2D cross- section represents a 3D body.

#### B

**BARLOW POINTS**

The set of Gauss integration points that give the best estimates of the stress for an element. For triangles and tetrahedra these are the full Gauss integration points. For quadrilateral and brick elements they are the reduced Gauss points.

**BASIS SPACE**

When an element is being constructed it is derived from a simple regular shape in non-dimensional coordinates. The coordinates used to define the simple shape form the basis space. In its basis space a general quadrilateral is a 2×2 square and a general triangle is an isosceles triangle with unit side lengths.

**BEAM ELEMENT**

A line element that has both translational and rotational degrees of freedom. It represents both membrane and bending actions.

**BENDING**

Bending behavior is where the strains vary linearly from the centerline of a beam or center surface of a plate or shell.There is zero strain on the centerline for pure bending. Plane sections are assumed to remain plane. If the stresses are constant normal to the centerline then this is called membrane behavior.

**BENDING STRESS**

A compressive and/or tensile stress resulting from the application of a nonaxial force to a structural member.

**BODY FORCE VECTOR**

Mechanical loadings within the interior of the volume, typically inertia loadings in a stiffness analysis.

**BOUNDARY ELEMENT BOUNDARY INTEGRAL**

A method of solving differential equations by taking exact solutions to the field equations loaded by a point source and then finding the strengths of sources distributed around the boundary of the body required to satisfy the boundary conditions on the body.

**BUBBLE FUNCTIONS**

Element shape functions that are zero along the edges of the element. They are non-zero within the interior of the element.

**BUCKLING (SNAP THROUGH)**

The situation where the elastic stiffness of the structure is cancelled by the effects of compressive stress within the structure. If the effect of this causes the structure to suddenly displace a large amount in a direction normal to the load direction then it is classical bifurcation buckling. If there is a sudden large movement in the direction of the

loading it is snap through buckling.

#### C

**CENTRAL DIFFERENCE METHOD**

A method for numerically integrating second order dynamic equations of motion. It is widely used as a technique for solving non-linear dynamic problems.

**CHARACTERISTIC VALUE**

Same as the eigenvalue.

**CHARACTERISTIC VECTOR**

Same as the eigenvector.

**CHOLESKY FACTORISATION (SKYLINE)**

A method of solving a set of simultaneous equations that is especially well suited to the finite element method. It is sometimes called a skyline solution. Choose to optimize the profile o f the matrix if a renumbering scheme is used.

**COEFFICIENT OF VISCOUS DAMPING**

The system parameter relating force to velocity.

**COLUMN VECTOR (COLUMN MATRIX)**

An nx1 matrix written as a vertical string of numbers. It is the transpose of a row vector.

**COMPATIBILITY EQUATIONS**

Compatibility is satisfied if a field variable, typically the structural displacement, which is continuous before loading is continuous after loading. For linear problems the equations of compatibility must be satisfied. Nonlinearity in or non-satisfaction of, the compatibility equations leads to cracks and gaps in the structure. For finite element solutions compatibility of displacement is maintained within the element and across element

boundaries for the most reliable forms of solution.

**COMPATIBILITY OF STRAINS**

Compatibility of strain is satisfied if strains that are continuous before loading are continuous after.

**COMPLETE DISPLACEMENT FIELD**

When the functions interpolating the field variable (typically the displacements) form a complete n^{th} order polynomial in all directions.

**COMPLEX EIGENVALUES**

The eigenvectors of a damped system. For proportionally damped systems they are the same as the undamped eigenvectors. For non-proportionally damped systems with damping in all modes less th an critical they are complex numbers and occur as complex conjugate pairs.

**COMPLEX EIGENVECTORS**

The eigenvalues of any damped system. If the damping is less than critical they will occur as complex conjugate pairs even for proportionally damped systems. The real part of the complex eigenvalue is a measure of the damping in the mode and should always be negative. The imaginary part is a measure of the resonant frequency.

**COMPOSITE MATERIAL**

A material that is made up of discrete components, typically a carbon-epoxy composite material or a glass-fiber material. Layered material and foam materials are also forms of composite materials.

**COMPUTATIONAL FLUID DYNAMICS (CFD)**

A computer-based numerical study of turbulent fluid flow using approximate methods such as the finite element method, the fine difference method, the boundary element method, the finite volume methods, and so on.

**CONDENSATION STATIC CONDENSATION MODAL CONDENSATION**

The reduction of the size of a problem by eliminating (condensing out) some degrees of freedom. For static condensation the elimination process is based upon static considerations alone. In more general condensation it can include other effects, typically model condensation includes both static and dynamic effects.

**CONDITION NUMBER**

The ratio of the highest eigenvalue to the lowest eigenvalue of a matrix. The exponent of this number gives a measure of the number of digits required in the computation to maintain numerical accuracy. The higher the condition number the more chance of numerical error and the slower the rate of convergence for iterative solutions.

**CONDITIONAL STABILITY UNCONDITIONAL STABILITY**

Any scheme for numerically integrating dynamic equations of motion in a step-by- step form is conditionally stable if there is a maximum time step value that can be used. It is unconditionally stable (but not necessarily accurate) if any length of time step can be used.

**CONGRUENT TRANSFORMATION**

A transformation of the coordinate system of the problem that preserves the symmetry of the system m atrices.

**CONJUGATE GRADIENT METHOD**

A method for solving simultaneous equations iteratively. It is closely related to the Lanczos method for finding the first few eigenvalues and eigenvectors of a set of equations.

**CONSISTENT DISPLACEMENTS AND FORCES**

The displacements and forces act at the same point and in the same direction so that the sum of their products give a work quantity. If consistent displacements and forces are used the resulting stiffness and mass matrices are symmetric.

**CONSTANT STRAIN CONSTANTSTRESS**

For structural analysis an element must be able to reproduce a state of constant stress and strain under a suitable loading to ensure that it will converge to the correct solution. This is tested for using the patch test.

**CONSTITUTIVE RELATIONSHIPS**

The equations defining the material behavior for an infinitesimal volume of material. For structures these are the stress -strain laws and include Hookes law for elasticity and the Prandle-Reuss equations for plasticity.

**CONSTRAINT EQUATIONS (MULTI POINT CONSTRAINTS)**

If one group of variables can be defined in terms of another group then the relationship between the two are constraint equations. Typically the displacements on the face of an element can be constrained to remain plane but the plane itself can move.

**CONSTRAINTS**

Known values of, or relationships between, the displacements in the coordinate system.

**CONTACT PROBLEMS**

A contact problem occurs when two bodies that are originally apart can come together, or two bodies that are originally connected can separate.

**CONTINUOUS MASS MODELS**

The system mass is distributed between the degrees of freedom. The mass matrix is not diagonal.

**CONTINUOUS MODELS**

The model is defined in terms of partial differential equations rather than in finite degree of freedom matrix form.

**CONTOUR PLOTTING**

A graphical representation of the variation of a field variable over a surface, such as stress, displacement, or temperature. A contour line is a line of constant value for the variable. A contour band is an area of a single color for values of the variable within two limit values.

**CONVERGENCE REQUIREMENTS**

For a structural finite element to converge as the mesh is refined it must be able to represent a state of constant stress and strain free rigid body movements exactly. There are equivalent requirements for other problem types.

**CONVOLUTION INTEGRAL (DUHAMEL INTEGRAL)**

The integral relating the dynamic displacement response of the structure at any time t to the forces applied before this time.

**COORDINATE SYSTEM**

The set of displacements used to define the degrees of freedom of the system.

**CORRESPONDING FORCES AND DISPLACEMENTS**

A force and a displacement are said to correspond if they act at the same point and in the same direction. Forces and translational displacements can correspond as can moments and rotations. Corresponding forces and displacements can be multiplied together to give a work quantity. Using corresponding forces and displacements will always lead to a symmetric stiffness matrix.

**CRACK ELEMENT (CRACK TIP ELEMENT)**

An element that includes special functions to model the stress field at the tip of a crack. This is commonly achieved by using quadratic elements with mid side nodes at the quarter chord points.

**CRACK PROPAGATION (FRACTURE MECHANICS)**

The process by which a crack can propagate through a structure. It is commonly assumed that a crack initiates when a critical value of stress or strain is reached and it propagates if it can release more than a critical amount of energy by the crack opening.

**CRANK-NICHOLSON SCHEME**

A method for numerically integrating first order dynamic equations of motion. It is widely used as a technique for solving thermal transient problems.

**CRITICAL ENERGY RELEASE**

This is a material property defining the minimum energy that a propagating crack must release in order for it to propagate. Three critical energies, or modes of crack propagation, have been identified. Mode 1 is the two surfaces of the crack moving apart. Mode 2 is where the two surfaces slide from front to back. Mode 3 is where the to surfaces slide sideways.

**CRITICALLY DAMPED SYSTEM CRITICAL DAMPING**

The dividing line between under damped and over damped systems where the equation of motion has a damping value that is equal to the critical damping.

**CYCLIC SYMMETRY**

A generalization of axisymmetry. The structure is composed of a series of identical sectors that are arranged circumferentially to form a ring. A turbine disc with blades attached is atypical example.

###### REFERENCES

http://www.cae.tntech.edu/~chriswilson/FEA/ANSYS/

www.nafems.org

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