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

#### D

**DAMPED EIGENVALUES**

Same as complex eigenvalues.

**DAMPED EIGENVECTORS**

Same as complex eigenvectors.

**DAMPED NATURAL FREQUENCY**

The frequency at which the damped system vibrates naturally when only an initial disturbance is applied.

**DAMPING**

Any mechanism that dissipates energy in a vibrating system.

**DAMPING FACTOR (DECAY FACTOR)**

The damping factor is the ratio of the actual damping to the critical damping. It is often specified as a percentage. If the damping factor is less than one then the system can undergo free vibrations. The free vibrations will decay to zero with time. If the damping factor is greater than one then the decay is exponential and no vibrations occur. For most structures the damping factor is very small.

**DEGENERATE ELEMENTS**

Elements that are defined as one shape in the basis space but they are a simpler shape in the real space. A quadrilateral can degenerate into a triangle. A brick element can degenerate into a wedge, a pyramid or a tetrahedron. Degenerate elements should be avoided in practice.

**DEGREES OF FREEDOM**

The number of equations of equilibrium for the system. In dynamics, the number of displacement quantities which must be considered in order to represent the effects of all of the significant inertia forces. Degrees of freedom define the ability of a given node to move in any direction in space.

There are six types of DOF for any given node:

3 possible translations (one each in the X,Y and Z directions) and 3 possible rotations (one rotation about each of the X,Y, and X axes).

DOF are defined and restricted by the elements and constraints associated with each node.

**DET(J) DET J**

The Jacobian matrix is used to relate derivatives in the basis space to the real space. The determinant of the Jacobian – det(j) -is a measure of the distortion of the element when mapping from the basis to the real space.

**DETERMINISTIC ANALYSIS**

The applied loading is a known function of time.

**DEVIATORIC STRESS STRESS DEVIATORS**

A measure of stress where the hydrostatic stress has been subtracted from the actual stress. Material failures that are flow failures (plasticity and creep) fail independently of the hydrostatic stress. The failure is a function of the deviatoric stress.

**DIAGONAL DECAY**

When a matrix is factorized into a triangular form the ratio of a diagonal term in the factorized matrix to the corresponding term in the original matrix decreases in size as one moves down the diagonal. If the ratio goes to zero the matrix is singular and if it is negative the matrix is not positive definite. The diagonal decay can be used as an approximate estimate of the condition number of the matrix.

**DIAGONAL GENERALIZED MATRIX**

The eigenvectors of a system can be used to define a coordinate transformation such that, in these generalized coordinates the coefficient matrices (typically mass and stiffness) are diagonal.

**DIE-AWAY LENGTH**

If there is a stress concentration in a structure the high stress will reduce rapidly with distance from the peak value. The distance over which it drops to some small value isÂ called the die-away length. A fine mesh is required over this die-away length for accurate stress results.

**DIRECT INTEGRATION**

The name for various techniques for numerically integrating equations of motion. These are either implicit or explicit methods and include central difference, Crank-Nicholson, Runge-Kutta, Newmark beta and Wilson theta.

**DIRECTION COSINES**

The cosines of the angles a vector makes with the global x,y,z axes.

**DISCRETE PARAMETER MODELS (DISCRETISED APPROACH)**

The model is defined in terms of an ordinary differential equation and the system has a finite number of degrees of freedom.

**DISCRETIZATION**

The process of dividing geometry into smaller pieces (finite elements) to prepare for analysis, i.e. Meshing.

**DISPLACEMENT METHOD (DISPLACEMENT SOLUTION)**

A form of discrete parameter model where the displacements of the system are the basic unknowns.

**DISPLACEMENT**

The distance, translational and rotational, that a node travels from its initial position to its post-analysis position. The total displacement is represented by components in each of the 3 translational directions and the 3 rotational directions.

**DISPLACEMENT PLOTS**

Plots showing the deformed shape of the structure. For linear small deflection problems the displacements are usually multiplied by a magnifying factor before plotting the deformed shape.

**DISPLACEMENT VECTOR**

The nodal displacements written as a column vector.

**DISSIMILAR SHAPE FUNCTIONS INCOMPATIBLE SHAPE FUNCTIONS**

If two connecting elements have different shape functions along the connection line they are said to be incompatible. This should be avoided since convergence to the correct solution cannot be guarantied.

**DISTORTION ELEMENT DISTORTION**

Elements are defined as simple shapes in the basis space, quadrilaterals are square, triangles are isosoles triangles. If they are not this shape in the real space they are said to be distorted. Too much distortion can lead to errors in the solution

**DRUCKER-PRAGER EQUIVALENT STRESSES**

An equivalent stress measure for friction materials (typically sand). The effect of hydrostatic stress is included in the equivalent stress.

**DYNAMIC ANALYSIS**

An analysis that includes the effect of the variables changing with time as well as space.

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**DYNAMIC FLEXIBILITY MATRIX**

The factor relating the steady state displacement response of a system to a sinusoidal force input. It is the same as the recep tance.

**DYNAMIC MODELLING**

A modeling process where consideration as to time effects in addition to spatial effects are included. A dynamic model can be the same as a static model or it can differ significantly depending upon the nature of the problem.

**DYNAMIC RESPONSE**

The time dependent response of a dynamic system in terms of its displacement, velocity or acceleration at any given point of the system.

**DYNAMIC STIFFNESS MATRIX**

If the structure is vibrating steadily at a frequency w then the dynamic stiffness is (K+iwC w2M) It is the inverse of the dynamic flexibility matrix.

**DYNAMIC STRESSES**

Stresses that vary with time and space.

**DYNAMIC SUBSTRUCTURING**

Special forms of substructuring used within a dynamic analysis. Dynamic substructuring is always approximate and causes some loss of accuracy in the dynamic solution.

#### E

**EIGENVALUE PROBLEM**

Problems that require calculation of eigenvalues and eigenvectors for their solution. Typically solving free vibration problems or finding buckling loads.

**EIGENVALUES LATENT ROOTS CHARACTERISTIC VALUES**

The roots of the characteristic equation of the system. If a system has n equations of motion then it has n eigenvalues. The square root of the eigenvalues are the resonant frequencies. These are the frequencies that the structure will vibrate at if given some initial disturbance with no other forcing. There are other problems that require the solution of the eigenvalue problem, the buckling loads of a structure are eigenvalues. Latent roots and

characteristic values are synonyms for eigenvalues.

**EIGENVECTORS LATENT VECTORS NORMAL MODES**

The displacement shape that corresponds to the eigenvalues. If the structure is excited at a resonant frequency then the shape that it adopts is the mode shape corresponding to the eigenvalue. Latent vectors and normal modes are the same as eigenvectors.

**ELASTIC FOUNDATION**

If a structure is sitting on a flexible foundation the supports are treated as a continuous elastic foundation. The elastic foundation can have a significant effect upon the structural response.

**ELASTIC STIFFNESS**

If the relationship between loads and displacements is linear then the problem is elastic. For a multi-degree of freedom system the forces and displacements are related by the elastic stiffness matrix.

**ELECTRIC FIELDS**

Electro-magnetic and electro-static problems form electric field problems.

**ELEMENT **

In the finite element method the geometry is divided up into elements, much like basic building blocks. Each element has nodes associated with it. The behavior of the element is defined in terms of the freedoms at the nodes.

**ELEMENT ASSEMBLY**

Individual element matrices have to be assembled into the complete stiffness matrix. This is basically a process of summing the element matrices. This summation has to be of the correct form. For the stiffness method the summation is based upon the fact that element displacements at common nodes must be the same.

**ELEMENT STRAINS ELEMENT STRESSES**

Stresses and strains within elements are usually defined at the Gauss points (ideally at the Barlow points) and the node points. The most accurate estimates are at the reduced Gauss points (more specifically the Barlow points). Stresses and strains are usually calculated here and extrapolated to the node points.

**ENERGY METHODS HAMILTONS PRINCIPLE**

Methods for defining equations of equilibrium and compatibility through consideration of possible variations of the energies of the system. The general form is Hamiltons principle and sub-sets of this are the principle of virtual work including the principle of virtual displacements (PVD) and the principle of virtual forces (PVF).

**ENGINEERING NORMALIZATION MATHEMATICAL NORMALIZATION**

Each eigenvector (mode shape or normal mode) can be multiplied by an arbitrary constant and still satisfy the eigenvalue equation. Various methods of scaling the eigenvector are used Engineering normalization -The vector is scaled so that the largest absolute value of any term in the eigenvector is unity. This is useful for inspecting printed tables of eigenvectors. Mathematical normalization -The vector is scaled so that the diagonal

modal mass matrix is the unit matrix. The diagonal modal stiffness matrix is the system eigenvalues. This is useful for response calculations.

**EQUILIBRIUM EQUATIONS**

Internal forces and external forces must balance. At the infinitesimal level the stresses and the body forces must balance. The equations of equilibrium define these force balance conditions.

**EQUILIBRIUM FINITE ELEMENTS**

Most of the current finite elements used for structural analysis are defined by assuming displacement variations over the element. An alternative approach assumes the stress variation over the element. This leads to equilibrium finite elements.

**EQUIVALENT MATERIAL PROPERTIES**

Equivalent material properties are defined where real material properties are smeared over the volume of the element. Typically, for composite materials the discrete fiber and matrix material properties are smeared to give average equivalent material properties.

**EQUIVALENT STRESS**

A three dimensional solid has six stress components. If material properties have been found experimentally by a uniaxial stress test then the real stress system is related to this by combining the six stress components to a single equivalent stress. There are various forms of equivalent stress for different situations. Common ones are Tresca, Von-Mises, Mohr-Coulomb and Drucker-Prager.

**ERGODIC PROCESS**

A random process where any one-sample record has the same characteristics as any other record.

**EULERIAN METHOD LAGRANGIAN METHOD**

For non-linear large deflection problems the equations can be defined in various ways. If the material is flowing though a fixed grid the equations are defined in Eulerian coordinates. Here the volume of the element is constant but the mass in the element can change. If the grid moves with the body then the equations are defined in Lagrangian coordinates. Here the mass in the element is fixed but the volume changes.

**EXACT SOLUTIONS**

Solutions that satisfy the differential equations and the associated boundary conditions exactly. There are very few such solutions and they are for relatively simple geometries and loadings.

**EXPLICIT METHODS IMPLICIT METHODS**

These are methods for integrating equations of motion. Explicit methods can deal with highly non-linear systems but need small steps. Implicit methods can deal with mildly nonlinear problems but with large steps.

**EXTRAPOLATION INTERPOLATION**

The process of estimating a value of a variable from a tabulated set of values. For interpolation values inside the table are estimated. For extrapolation values outside the table are estimated. Interpolation is generally accurate and extrapolation is only accurate for values slightly outside the table. It becomes very inaccurate for other cases.

#### F

**FACETED GEOMETRY**

If a curved line or surface is modeled by straight lines or flat surfaces then the modeling is said to produce a faceted geometry.

**FAST FOURIER TRANSFORM**

A method for calculating Fourier transforms that is computationally very efficient.

**FIELD PROBLEMS**

Problems that can be defined by a set of partial differential equations are field problems. Any such problem can be solved approximately by the finite element method.

**FINITE DIFFERENCES**

A numerical method for solving partial differential equations by expressing them in a difference form rather than an integral form. Finite difference methods are very similar to finite element methods and in some cases are identical.

**FINITE ELEMENT MODELING (FEM)**

The process of setting up a model for analysis, typically involving graphical generation of the model geometry, meshing it into finite elements, defining material properties, and applying loads and boundary conditions.

**FINITE VOLUME METHODS**

A technique related to the finite element method. The equations are integrated approximately using t he weighted residual method, but a different form of weighting function is used from that in the finite element method. For the finite element method the Galerkin form of the weighted residual method is used.

**FIXED BOUNDARY CONDITIONS**

All degrees of freedom are restrained for this condition. The nodes on the fixed boundary can not move: translation or rotation.

**FLEXIBILITY MATRIX FORCE METHOD**

The conventional form of the finite element treats the displacements as unknowns, which leads to a stiffness matrix form. Alternative methods treating the stresses (internal forces) as unknowns lead to force methods with an associated flexibility matrix. The inverse of the stiffness matrix is the flexibility matrix.

**FORCED RESPONSE**

The dynamic motion results from a time varying forcing function.

**FORCING FUNCTIONS**

The dynamic forces that are applied to the system.

**FOURIER EXPANSIONS FOURIER SERIES**

Functions that repeat themselves in a regular manner can be expanded in terms of a Fourier series.

**FOURIER TRANSFORM**

A method for finding the frequency content of a time varying signal. If the signal is periodic it gives the same result as the Fourier series.

**FOURIER TRANSFORM PAIR**

The Fourier transform and its inverse which, together, allow the complete system to be transformed freely in either direction between the time domain and the frequency domain.

**FRAMEWORK ANALYSIS**

If a structure is idealized as a series interconnected line elements then this forms a framework analysis model. If the connections between the line elements a re pins then it is a pin-jointed framework analysis. If the joints are rigid then the lines must be beam elements.

**FREE VIBRATION**

The dynamic motion which results from specified initial conditions. The forcing function is zero.

**FREQUENCY DOMAIN**

The structures forcing function and the consequent response is defined in terms of their frequency content. The inverse Fourier transform of the frequency domain gives the corresponding quantity in the time domain.

**FRONTAL SOLUTION WAVEFRONT SOLUTION**

A form of solving the finite element equations using Gauss elimination that is very efficient for the finite element form of equations.

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###### REFERENCES

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

www.nafems.org

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