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

### O

**OPTIMAL SAMPLING POINTS**

The minimum number of Gauss points required to integrate an element matrix. Also the Gauss points at which the stresses are most accurate (see reduced Gauss points).

**OVER DAMPED SYSTEM**

A system which has an equation of motion where the damping is greater than critical. It has an exponentially decaying, non-oscillatory impulse response.

**OVERSTIFF SOLUTIONS**

Lower bound solutions. These are associated with the assumed displacement method.

### P

**PARAMETRIC STUDIES PILOT STUDIES**

Initial studies conducted on small -simplified models to determine the important parameters in the solution of a problem. These are often used to determine the basic mesh density required.

**PARTICIPATION FACTOR**

The fraction of the mass that is active for a given mode with a given distribution of dynamic loads. Often this is only defined for the specific load case of inertia (seismic) loads.

**PATCH TEST**

A test to prove that a mesh of distorted elements can represent constant stress situations and strain free rigid body motions (i.e. the mesh convergence requirements) exactly.

**PERIODIC RESPONSE FORCE**

A response (force) that regularly repeats itself exactly.

**PHASE ANGLE**

The ratio of the in phase component of a signal to its out of phase component gives the tangent of the phase angle of the signal relative to some reference.

**PLANE STRAIN PLANE STRESS**

A two dimensional analysis is plane stress if the stress in the third direction is assumed zero. This is valid if the dimension of the body in this direction is very small, e.g. a thin plate. A two dimensional analysis is plane strain if the strain in the third direction is assumed zero. This is valid if the dimension of the body in this direction is very large, e.g. a cross- sectional slice of a long body.

**PLATE BENDING ELEMENTS**

Two-dimensional shell elements where the in plane behavior of the element is ignored. Only the out of plane bending is considered.

**POISSONS RATIO**

The material property in Hooke s law relating strain in one direction arising from a stress in a perpendicular direction to this.

**POST ANALYSIS CHECKS**

Checks that can be made on the results after the analysis. For a stress analysis these could include how well stress free boundary conditions have been satisfied or how continuous stresses are across elements.

**POST-PROCESSING**

The interrogation of the results after the analysis phase. This is usually done with a combination of graphics and numerics.

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**POTENTIAL ENERGY**

The energy associated with the static behavior of a system. For a structure this is the strain energy.

**POTENTIAL FLOW**

Fluid flow problems where the flow can be represented by a scalar potential function.

**POWER METHOD**

A method for finding the lowest or the highest eigenvalue of a system.

**PRANDTL-REUSS EQUATIONS**

The equations relating an increment of stress to an increment of plastic strain for a metal undergoing plastic flow.

**PREPROCESSING**

The process of preparing finite element input data involving model creation, mesh generation,material definition, and load and boundary condition application.

**PRIMARY COMPONENT**

Those parts of the structure that are of direct interest for the analysis. Other parts are secondary components.

**PRINCIPAL CURVATURE**

The maximum and minimum radii of curvature at a point.

**PRINCIPAL STRESSES**

The maximum direct stress values at a point. They are the eigenvalues of the stress tensor.

**PROFILE**

The profile of a symmetric matrix is the sum of the number of terms in the lower (or upper) triangle of the matrix ignoring the leading zeros in each row. Embedded zeros are included in the count. It gives a measure of the work required to factorize the matrix when using the Cholesky solution. Node renumbering minimizes it.

**PROPORTIONAL DAMPING**

A damping matrix that is a linear combination of the mass and stiffness matrices. The eigenvectors of a proportionally damped system are identical to those of the undamped system.

**P-METHOD**

A method of finite element analysis that uses P- convergence to iteratively minimize the error of analysis.

**QR METHOD**

A technique for finding eigenvalues. This is currently the most stable method for finding eigenvalues but it is restricted in the size of problem that it can solve.

**RANDOM VIBRATIONS**

The applied loading is only known in terms of its statistical properties. The loading is nondeterministic in that its value is not known exactly at any time but its mean, mean square, variance and other statistical quantities are known.

### R

**RANK DEFICIENCY**

A measure of how singular a matrix is.

**RAYLEIGH DAMPING**

Damping that is proportional to a linear combination of the stiffness and mass. This assumption has no physical basis but it is mathematically convenient to approximate low damping in this way when exact damping values are not known.

**RAYLELGH QUOTIENT**

The ratio of stiffness times displacement squared (2*strain energy) to mass times

displacement squared. The minimum values of the Rayleigh quotient are the eigenvalues.

**REACTION FORCES**

The forces generated at support points when a structure is loaded.

**REFERENCE TEMPERATURE**

The reference temperature defines the temperature at which strain in the design does not

result from thermal expansion or contraction. For many situations, reference temperature

is adequately defined as room temperature. Define reference te mperature in the

properties of an environment.

**RECEPTANCE**

The ratio of the steady state displacement response to the value of the forcing function for

a sinusoidal excitation. It is the same as the dynamic flexibility.

**REDUCED INTEGRATION**

If an element requires an l*m*n Gauss rule to integrate the element matrix exactly then (l-1)*(m-1)*(n-1) is the reduced integration rule. For many elements the stresses are most

accurate at the reduced integration points. For some elements the matrices are best

evaluated by use of the reduced integration points. Use of reduced integration for

integrating the elements can lead to zero energy and hour glassing modes.

**RESPONSE SPECTRUM METHOD**

A method for characterizing a dynamic transient forcing function and the associated

solution technique. It is used for seismic and shock type loads.

**RESTARTS CHECKPOINTS**

The process whereby an analysis can be stopped part way through and the analysis restarted at a later time.

**RIGID BODY DEFORMATIONS**

A non-zero displacement pattern that h as zero strain energy associate with it.

**RIGID BODY DISPLACEMENT**

A non-zero displacement pattern that has zero strain energy associate with it.

**RIGID BODY MODES**

If a displaced shape does not give rise to any strain energy in the structure then this a rigid body mode. A general three-dimensional unsupported structure has 6 rigid body modes, 3 translation and 3 rotation.

**RIGID LINKS RIGID OFFSETS**

This is a connection between two non-coincident nodes assuming that the connection is infinitely stiff. This allows the degrees of freedom at one of the nodes (the slave node) to be deleted from the system. It is a form of multi-point constraint.

**ROUNDOFF ERROR**

Computers have a fixed word length and hence only hold numbers to a certain number of significant figures. If two close numbers are subtracted one from another then the result loses the first set of significant figures and hence loses accuracy. This is round off error.

**ROW VECTOR ROW MATRIX**

A 1xn matrix written as a horizontal string of numbers. It is the transpose of a column vector.

### S

**SCALARS VECTORS**

Quantities that have no direction associated with them, e.g. temperatures. Scalar problems only have one degree of freedom at a node. Vector quantities have a direction associated with them, e.g. displacements. Vector problems have more than one degree of freedom at a node.

**SECANT STIFFNESS**

The stiffness defined by the slope of the line from the origin to the current point of interest on a load/deflection curve.

**SECONDARY COMPONENTS**

Components of a structure not of direct interest but they may have some influence of the behavior of the part of the structure that is of interest (the primary component) and have to be included in the analysis in some approximate form.

**SEEPAGE FLOW**

Flows in porous materials

**SEISMIC ANALYSIS**

The calculation of the dynamic displacement and stress response arising from earthquake excitations.

**SELECTED REDUCED INTEGRATION**

A form of Gaussian quadrature where different sets of Gauss points are used for different strain components.

**SELF ADJOINT EQUATIONS**

A form of matrix products that preserves symmetry of equations. The product A*B*A(transpose) is self -adjoint if the matrix B is symmetric. The result of the product will be symmetric for any form of A that is of a size compatible with B. This form o f equation occurs regularly within the finite element method. Typically it means that for a structural analysis the stiffness (and mass) matrices for any element or element assembly will be

symmetric.

**SELF EQUILIBRATING LOADS**

A load set is self -equilibrating if all of its resultants are zero. Both translation and moment resultants are zero.

**SEMI-LOOF ELEMENT**

A form of thick shell element.

**SHAKEDOWN**

If a structure is loaded cyclically and initially undergoes some plastic deformation then it is said to shakedown if the behavior is entirely elastic after a small number of load cycles.

**SIMPSONS RULE**

A method for numerically integrating a function.

**SIMULTANEOUS VECTOR ITERATION**

A method for finding the first few eigenvalues and eigenvectors of a finite element system. This is also known as subspace vector iteration.

**SINGLE DEGREE OF FREEDOM**

The system is defined by a single force/displacement equation.

**SINGLE POINT CONSTRAINT**

Where the constraint is unique to a single node point.

**SINGULAR MATRIX**

A square matrix that cannot be inverted.

**SKEW DISTORTION (ANGULAR DISTORTION)**

A measure of the angular distortion arising between two vectors that are at right angles in the basis space when these are mapped to the real coordinate space. If this angle approaches zero the element becomes ill-conditioned.

**SOLID ELEMENTS**

Three dimensional continuum elements.

**SOLUTION DIAGNOSTICS**

Messages that are generated as the finite element solution progresses. These should always be checked for relevance but the are often only provided for information purposes

**SOLUTION EFFICIENCY**

A comparative measure between two solutions of a given problem defining which is the ‘best’. The measures can include accuracy, time of solution, memory requirements and disc storage space.

**SPARSE MATRIX METHODS**

Solution methods that exploit the sparse nature of finite element equations. Such methods include the frontal solution and Cholesky (skyline) factorization for direct solutions, conjugate gradient methods for iterative solutions and the Lanczos method and subspace iteration (simultaneous vector iteration) for eigenvalue solutions.

**SPECTRAL DENSITY**

The Fourier transform of the correlation function. In random vibrations it gives a measure of the significant frequency content in a system. White noise has a constant spectral density for all frequencies.

**SPLINE CURVES**

A curve fitting technique that preserves zero, first and second derivative continuity across segment boundaries.

**SPURIOUS CRACKS**

Cracks that appear in a mesh when the elements are not correctly connected together. This is usually an error in the mesh generation process.

**STATIC ANALYSIS**

Analysis of stresses and displacements in a structure when the applied loads do not vary with time.

**STATICALLY DETERMINATE STRUCTURE**

A structure where all of the unknowns can be found from equilibrium considerations alone.

**STATICALLY EQUIVALENT LOADS**

Equivalent nodal loads that have the same equilibrium resultants as the applied loads but do not necessarily do the same work as the applied loads.

**STATICALLY INDETERMINATE STRUCTURE REDUNDANT**

A structure where all of the unknowns can not be found from equilibrium considerations alone. The compatibility equations must also be used. In this case the structure is said to be redundant.

**STATIONARY RANDOM EXCITATION**

A force or response that is random but its statistical characteristics do not vary with time.

**STEADY-STATE HEAT TRANSFER**

Determination of the temperature distribution of a mechanical part having reached thermal equilibrium with the environmental conditions. There are no time varying changes in the resulting temperatures.

**STEADY STATE RESPONSE**

The response of the system to a periodic forcing function when all of the transient components of the response have become insignificant.

**STEP-BY-STEP INTEGRATION**

Methods of numerically integrating time varying equations of motion. These methods can be either explicit or implicit.

**STIFFNESS**

A set of values which represent the rigidity or softness of a particular element. Stiffness is determined by material type and geometry.

**STIFFNESS MATRIX**

The parameter(s) that relate the displacement(s) to the force(s). For a discrete parameter multi degree of freedom model this is usually given as a stiffness matrix.

**STRAIN**

A dimensionless quantity calculated as the ratio of deformation to the original size of the body.

**STRAIN ENERGY**

The energy stored in the system by the stiffness when it is displaced from its equilibrium position.

**STRESS**

The intensity of internal forces in a body (force per unit area) acting on a plane within the material of the body is called the stress on that plane.

**STRESS ANALYSIS**

The computation of stresses and displacements due to applied loads. The analysis may be elastic, inelastic, time dependent or dynamic.

**STRESS AVERAGING STRESS SMOOTHING**

The process of filtering the raw finite element stress results to obtain the most realistic estimates of the true state of stress.

**STRESS CONCENTRATION**

A local area of the structure where the stresses are significantly higher than the general stress level. A fine mesh of elements is required in such regions if accurate estimates of the stress concentration values are required.

**STRESS CONTOUR PLOT**

A plot of a stress component by a series of color filled contours representing regions of equal stress.

**STRESS DISCONTINUITIES STRESS ERROR ESTIMATES**

Lines along which the stresses are discontinuous. If the geometry or loading changes abruptly along a line then the true stress can be discontinuous. In a finite element solution the element assumptions means that the stresses will generally be discontinuous across element boundaries. The degree of discontinuity can then be used to form an estimate of the error in the stress within the finite element calculation.

**STRESS EXTRAPOLATION**

The process of taking the stress results at the optimum sampling points for an element and extrapolating these to the element node points.

**STRESS INTENSITY FACTORS**

A measure of the importance of the stress at a sharp crack tip (where the actual stress values will be infinite) used to estimate if the crack will propagate.

**STRESS VECTOR STRESS TENSOR STRAIN VECTOR STRAIN TENSOR**

The stress (strain) vector is the components of stress (strain) written as a column vector. For a general three dimensional body this is a (6×1) matrix. The components of stress (strain) written in tensor form. For a general three dimensional body this forms a (3×3) matrix with the direct terms down the diagonal and the shear terms as the off-diagonals.

**STRESS-STRAIN LAW**

The material property behavior relating stress to strain. For a linear behavior this is Hookes law (linear elasticity). For elastic plastic behavior it is a combination of Hookes law and the Prandtl-Reuss equations.

**SUBSPACE VECTOR ITERATION**

A method for finding the first few eigenvalues and eigenvectors of a finite element system. This is also known as simultaneous vector iteration.

**SUBSTRUCTURING**

An efficient way of solving large finite element analysis problems by breaking the model into several parts or substructures, analyzing each one individually, and then combining them for the final results.

**SUBSTRUCTURING SUPER ELEMENT METHOD**

Substructuring is a form of equation solution method where the structure is split into a series of smaller structures -the substructures. These are solved to eliminate the internal freedoms and the complete problem solved by only assembling the freedoms on the common boundaries between the substructures. The intermediate solution where the internal freedoms of a substructure have been eliminated gives the super element matrix

for the substructure.

**SURFACE MODELING**

The geometric modeling technique in which the model is created in terms of its surfaces only without any volume definition.

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###### REFERENCES

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

www.nafems.org

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