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

### L

**LAGRANGE INTERPOLATION LAGRANGE SHAPE FUNCTIONS**

A method of interpolation over a volume by means of simple polynomials. This is the basis of most of the shape function definitions for elements.

**LAGRANGE MULTIPLIER TECHNIQUE**

A method for introducing constraints into an analysis where the effects of the constraint are represented in terms of the unknown Lagrange multiplying factors.

**LANCZOS METHOD**

A method for finding the first few eigenvalues and eigenvectors of a set of equations. It is very well suited to the form of equations generated by the finite element method. It is closely related to the method of conjugate gradients used for solving simultaneous equations iteratively.

**LEAST SQUARES FIT**

Minimization of the sum of the squares of the distances between a set of sample points and a smooth surface . The finite element method gives a solution that is a least squares fit to the equilibrium equations.

**LINEAR DEPENDENCE**

One or more rows (columns) of a matrix are linear combinations of the other rows (columns). This means that the matrix is singular.

**LINEAR ANALYSIS**

Analysis in which the displacements of the structure are linear functions of the applied loads.

**LINEAR SYSTEM**

When the coefficients of stiffness, mass and damping are all constant then the system is linear. Superposition can be used to solve the response equation.

**LOADINGS**

The loads applied to a structure that result in deflections and consequent strains and stresses.

**LOCAL STRESSES**

Areas of stress that are significantly different from (usually higher than) the general stress level.

**LOWER BOUND SOLUTION UPPER BOUND SOLUTION**

The assumed displacement form of the finite element solution gives a lower bound on the maximum displacements and strain energy (i.e. these are under estimated) for a given set of forces. This is the usual form of the finite element method. The assumed stress form of the finite element solution gives an upper bound on the maximum stresses and strain energy (i.e. these are over estimated) for a given set of displacements.

**LUMPED MASS MODEL**

When the coefficients of the mass matrix are combined to produce a diagonal matrix. The total mass and the position of the structures center of gravity are preserved.

### M

**MASS**

The constant(s) of proportionality relating the acceleration(s) to the force(s). For a discrete parameter multi degree of freedom model this is usually given as a mass matrix.

**MASS ELEMENT**

An element lumped at a node representing the effect of a concentrated mass in different coordinate directions.

**MASS MATRIX**

The matrix relating acceleration to forces in a dynamic analysis. This can often be approximated as a diagonal matrix with no significant loss of accuracy.

**MASTER FREEDOMS**

The freedoms chosen to control the structural response when using a Guyan reduction or sub structuring methods.

**MATERIAL LOSS FACTOR**

A measure of the damping inherent within a material when it is dynamically loaded.

**MATERIAL PROPERTIES**

The physical properties required to define the material behavior for analysis purposes. For stress analysis typical required material properties are Young’s modulus, Poisson’s ratio, density and coefficient of linear expansion. The material properties must have been obtained by experiment.

**MATERIAL STIFFNESS MATRIX MATERIAL FLEXIBILITY MATRIX**

The material stiffness matrix allows the stresses to be found from a given set of strains at a point. The material flexibility is the inverse of this, allowing the strains to be found from a given set of stresses. Both of these matrices must be symmetric and positive definite.

**MATRIX DISPLACEMENT METHOD**

A form (the standard form) of the finite element method where displacements are assumed over the element. This gives a lower bound solution.

**MATRIX FORCE METHOD**

A form of the finite element method where stresses (internal forces) are assumed over the element. This gives an upper bound solution.

**MATRIX INVERSE**

If matrix A times matrix B gives the unit matrix then A is the inverse of B (B is the inverse of A). A matrix has no inverse if it is singular.

**MATRIX NOTATION MATRIX ALGEBRA**

A form of notation for writing sets of equations in a compact manner. Matrix notation highlights the generality of various classes of problem formulation and solution. Matrix algebra can be easily programmed on a digital computer.

**MATRIX PRODUCTS**

Two matrices A and B can be multiplied together if A is of size (j*k) and B is of size (k*l). The resulting matrix is of size (j*l).

**MATRIX TRANSPOSE**

The process of interchanging rows and columns of a matrix so that the j’th column becomes the j’th row.

**MEAN SQUARE CONVERGENCE**

A measure of the rate of convergence of a solution process. A mean square convergence indicates a rapid rate of convergence.

**MEMBRANE**

Membrane behavior is where the strains are constant from the center line of a beam or center surface of a plate or shell. Plane sections are assumed to remain plane. A membrane line element only has stiffness along the line, it has zero stiffness normal to the line. A membrane plate has zero stiffness normal to the plate. This can cause zero energy (no force required) displacements in these normal directions. If the stresses vary linearly along the normal to the centerline then this is called bending behavior.

**MESH DENSITY MESH REFINEMENT**

The mesh density indicates the size of the elements in relation to the size of the body being analyzed. The mesh density need not be uniform all over the body There can be areas of mesh refinement (more dense meshes) in some parts of the body. Making the mesh finer is generally referred to as h -refinement. Making the element order higher is referred to as p -refinement.

**MESH GENERATION ELEMENT GENERATION**

The process of generating a mesh of elements over the structure. This is normally done automatically or semi-automatically.

**MESH SPECIFICATION**

The process of choosing and specifying a suitable mesh of elements for an analysis.

**MESH SUITABILITY**

The appropriate choice of element types and mesh density to give a solution to the required degree of accuracy.

**MINDLIN ELEMENTS**

A form of thick shell element.

**MOBILITY**

The ratio of the steady state velocity response to the value of the forcing function for a sinusoidal excitation.

**MODAL DAMPING**

The damping associated with the generalized displacements defined by the eigenvectors. Its value has no physical significance since the eigenvector contains an arbitrary normalizing factor.

**MODAL MASS**

The mass associated with the generalized displacements defined by the eigenvectors. Its value has no physical significance since the eigenvector contains an arbitrary normalizing factor but the ratio of modal stiffness to modal mass is always the eigenvalue.

**MODAL STIFFNESS**

The stiffness associated with the generalized displacements defined by the eigenvectors. Its value has no physical significance since the eigenvector contains an arbitrary normalizing factor but the ratio of modal stiffness to modal mass is always the eigenvalue.

**MODAL TESTING**

The experimental technique for measuring resonant frequencies (eigenvalues) and mode shapes (eigenvectors).

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**MODE PARTICIPATION FACTOR**

The generalized force in each modal equation of a dynamic system.

**MODE SHAPE**

Same as the e igenvector. The mode shape often refers to the measure mode, found from a modal test.

**MODELLING**

The process of idealizing a system and its loading to produce a numerical (finite element) model.

**MODIFIED NEWTON-RAPHSON**

A form of the Newton-Raphson process f or solving non-linear equations where the tangent stiffness matrix is held constant for some steps. It is suitable for mildly non-linear problems.

**MOHR COULOMB EQUIVALENT STRESS**

A form of equivalent stress that includes the effects of friction in granular (e.g. sand) materials.

**MULTI DEGREE OF FREEDOM**

The system is defined by more than one force/displacement equation.

**MULTI-POINT CONSTRAINTS**

Where the constraint is defined by a relationship between more than one displacement at different node points.

### N

**NATURAL FREQUENCY**

The frequency at which a structure will vibrate in the absence of any external forcing. If a model has n degrees of freedom then it has n natural frequencies. The eigenvalues of a dynamic system are the squares of the natural frequencies.

**NATURAL MODE**

Same as the eigenvector.

**NAVIER-STOKES EQUATIONS**

Partial differential equations defining the unsteady viscous flow of fluids.

**NEWMARK METHOD NEWMARK BETA METHOD**

An implicit solution method for integrating second order equations of motion. It can be made unconditionally stable.

**NEWTON COTES FORMULAE**

A family of methods for numerically integrating a function.

**NEWTON-RAPHSON NON-LINEAR SOLUTION**

A general technique for solving non-linear equations. If the function and its derivative are known at any point then the Newton-Raphson method is second order convergent.

**NODAL VALUES**

The value of variables at the node points. For a structure typical possible nodal values are force, displacement, temperature, velocity, x, y, and z.

**NODE NODES NODAL**

The point at which one element connects to another or the point where an element meets the model boundary. Nodes allow internal loads from one element to be transferred to another element. Element behavior is defined by the response at the nodes of the elements. Nodes are always at the corners of the element, higher order elements have nodes at mid-edge or other edge positions and some elements have nodes on faces or within the element volume. The behavior of the element is defined by the variables at the node. For a stiffness matrix the variables are the structural displacement, For a heat conduction analysis the nodal variable is the temperature. Other problems have other nodal variables.

**NON-CONFORMING ELEMENTS**

Elements that do not satisfy compatibility either within the element or across element boundaries or both. Such elements are not generally reliable although they might give very good solutions in some circumstances.

**NON-HOLONOMIC CONSTRAINTS**

Constraints that can only be defined at the level of infinitesimal displacements. They cannot be integrated to give global constraints.

**NON-LINEAR SYSTEM NON-LINEAR ANALYSIS**

When at least one of the coefficients of stiffness, mass or damping vary with displacement or time then the system is non-linear. Superposition cannot be used to solve the problem.

**NON-STATIONARY RANDOM**

A force or response that is random and its statistical properties vary with time.

**NON-STRUCTURAL MASS**

Mass that is present in the system and will affect the dynamic response but it is not a part of the structural mass (e.g. the payload).

**NORM**

A scalar measure of the magnitude of a vector or a matrix.

**NUMERICAL INTEGRATION**

The process of finding the approximate integral of a function by numerical sampling and summing. In the finite element method the element matrices are usually formed by the Gaussian quadrature form of numerical integration.

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###### REFERENCES

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

www.nafems.org

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