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

### T

**TEMPERATURE CONTOUR PLOTS**

A plot showing contour lines connecting points of equal temperature.

**TETRAHEDRON TETRAHEDRAL ELEMENT**

A three dimensional four sided solid element with triangular faces.

**THERMAL CAPACITY**

The material property defining the thermal inertia of a material. It relates the rate of change of temperature with time to heat flux.

**THERMAL CONDUCTIVITY**

The material property relating temperature gradient to heat flux.

**THERMAL LOADS**

The equivalent loads on a structure arising from thermal strains. These in turn arise from a temperature change.

**THERMAL STRAINS**

The components of strain arising from a change in temperature.

**THERMAL STRESS ANALYSIS**

The computation of stresses and displacements due to change in temperature.

**THIN SHELL ELEMENT THICK SHELL ELEMENT**

In a shell element the geometry is very much thinner in one direction than the other two. It can then be assumed stresses can only vary linearly at most in the thickness direction. If the through thickness shear strains can be taken as zero then a thin shell model is formed. This uses the Kirchoff shell theory If the transverse shear strains are not ignored then a thick shell model is formed. This uses the Mindlin shell theory. For the finite element method the thick shell theory generates the most reliable form of shell elements. There are two forms of such elements, the Mindlin shell and the Semi -Loof shell.

**TIMEDOMAIN**

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

**TRACE OF THE MATRIX**

The sum of the leading diagonal terms of the matrix.

**TRANSFINITE MAPPING**

A systematic method for generating element shape functions for irregular node distributions on an element.

**TRANSFORMATION METHOD**

Solution techniques that transform coordinate and force systems to generate a simpler form of solution. The eigenvectors can be used to transform coupled dynamic equations to a series of single degree of freedom equations.

**TRANSIENT FORCE**

A forcing function that varies for a short period of time and then settles to a constant value.

**TRANSIENT RESPONSE**

The component of the system response that does not repeat itself regularly with time.

**TRANSITION ELEMENT**

Special elements that have sides with different numbers of nodes. They are used to couple elements with different orders of interpolation, typically a transition element with two nodes on one edge and three on another is used to couple a 4 -node quad to an 8 -node quad.

**TRANSIENT HEAT TRANSFER**

Heat transfer problems in which temperature distribution varies as a function of time.

**TRIANGULAR ELEMENTS**

Two dimensional or surface elements that have three edges.

**TRUSS ELEMENT**

A one dimensional line element defined by two nodes resisting only axial loads.

### U

**ULTIMATE STRESS**

The failure stress (or equivalent stress) for the material.

**UNDAMPED NATURAL FREQUENCY**

The square root of the ratio of the stiffness to the mass (the square root of the eigenvalue). It is the frequency at which an undamped system vibrates naturally. A system with n degrees of freedom has n natural frequencies.

**UNDER DAMPED SYSTEM**

A system which has an equation of motion where the damping is less than critical. It has an oscillatory impulse response.

**UNIT MATRIX**

A diagonal matrix with unit values down the diagonal.

**UPDATED LAGRANGIAN TOTAL LAGRANGIAN**

The updated Lagrangian coordinate system is one where the stress directions are referred to the last known equilibrium state. The total Lagrangian coordinate system is one where the stress directions are referred to the initial geometry.

**UPWINDING IN FLUIDS**

A special form of weighting function used in viscous flow problems (solution to the NavierStokes equations) used in the weighted residual method to bias the results in the direction of the flow.

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### V

**VARIABLE BANDWIDTH (SKYLINE)**

A sparse matrix where the bandwidth is not constant. Some times called a skyline matrix.

**VELOCITY**

The first time derivative of the displacement.

**VIRTUAL CRACK EXTENSION CRACK PROPAGATION**

A technique for calculating the energy that would be released if a crack increased in size. This gives the energy release rate which can be compared to the critical energy release (a material property) to decide if a crack will propagate.

**VIRTUAL DISPLACEMENTS**

An arbitrary imaginary change of the system configuration consistent with its constraints.

**VIRTUAL WORK VIRTUAL DISPLACEMENTS VIRTUAL FORCES**

Techniques for using work arguments to establish equilibrium equations from compatibility equations (virtual displacements) and to establish compatibility equations from equilibrium (virtual forces).

**VISCOUS DAMPING**

The damping is viscous when the damping force is proportional to the velocity.

**VISCOUS DAMPING MATRIX**

The matrix relating a set of velocities to their corresponding velocities

**VOLUME DISTORTION VOLUMETRIC DISTORTION**

The distortion measured by the determinant of the Jacobian matrix, det j.

**VON MISES STRESS**

An “averaged” stress value calculated by adding the squares of the 3 component stresses (X, Y and Z directions) and taking the square root of their sums. This value allows for a quick method to locate probable problem areas with one plot.

**VON MISES EQUIVALENT STRESS TRESCA EQUIVALENT STRESS**

Equivalent stress measures to represent the maximum shear stress in a material. These are used to characterize flow failures (e.g. plasticity and creep). From test results the VonMises form seems more accurate but the Tresca form is easier to handle.

### W

**WAVE PROPAGATION**

The dynamic calculation involving the prediction of the history of stress and pressure waves in solids and fluids.

**WAVEFRONT (FRONT)**

The wavefront of a symmetric matrix is the maximum number of active nodes at any time during a frontal solution process. It is a measure of the time required to factorise the equations in a frontal solution. It is minimized be element renumbering.

**WEIGHTED RESIDUALS**

A technique for transforming a set of partial differential equations to a set of simultaneous equations so that the solution to the simultaneous equations satisfy the partial differential equations in a mean sense. The form used in the finite element method is the Galerkin process. This leads to identical equations to those from virtual work arguments.

**WHIRLING STABILITY**

The stability of rotating systems where centrifugal and Coriolis are also present.

**WHITE NOISE**

White noise has a constant spectral density for all frequencies.

**WILSON THETA METHOD**

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

**WORD LENGTH**

Within a digital computer a number is only held to a finite number of significant figures. A 32bit (single precision) word has about 7 significant figures. A 64bit (double precision) word has about 13 significant figures. All finite element calculations should be conducted in double precision.

### Y

**YOUNG’S MODULUS**

The material property relating a uniaxial stress to the corresponding strain.

### Z

**ZERO ENERGY MODES ZERO STIFFNESS MODES**

Non-zero patterns of displacements that have no energy associated with them. No forces are required to generate such modes, Rigid body motions are zero energy modes. Buckling modes at their buckling loads are zero energy modes. If the elements are not fully integrated they will have zero energy displacement modes. If a structure has one or more zero energy modes then the matrix is singular.

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###### REFERENCES

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

www.nafems.org