(A to Z) of Finite Element Analysis
GAP ELEMENT CONTACT ELEMENT
These are special forms of non-linear element that have a very high stiffness in compression and a low stiffness in tension. They are used to model contact conditions between surfaces. Most of these elements also contain a model for sliding friction between the contacting surfaces. Some gap elements are just line springs between points and
others are more general forms of quadrilateral or brick element elements. The line spring elements should only be used in meshes of first order finite elements.
GAUSS POINT EXTRAPOLATION GAUSS POINT STRESSES
Stresses calculated internally within the element at the Gauss integration points are called the Gauss point stresses. These stresses are usually more accurate at these points than the nodal points.
GAUSS POINTS GAUSS WEIGHTS
The Gauss points are the sample points used within the elements for the numerical integration of the matrices and loadings. They are also the points at which the stresses can be recovered. The Gauss weights are associated factors used in the numerical integration process. They represent the volume of influence of the Gauss points. The
positions of the Gauss points, together with the associated Gauss weights, are available in tables for integrations of polynomials of various orders.
A form of solving a large set of simultaneous equations. Within most finite element systems a form of Gaussian elimination forms the basic solution process.
GAUSSIAN INTEGRATION GAUSSIAN QUADRATURE
A form of numerically integrating functions that is especially efficient for integrating polynomials. The functions are evaluated at the Gauss points, multiplied by the Gauss weights and summed to give the integral.
A set of linearly independent displacement coordinates which are consistent with the constraints and are just sufficient to describe any arbitrary configuration of the system. Generalized coordinates are usually patterns of displacements, typically the system eigenvectors.
The mass associated with a generalized displacement.
The stiffness associated with a generalized displacement.
Various shape dependent properties of real structures, such as thickness, cross sectional area, sectional moments of inertia, centroid location and others that are applied as properties of finite elements.
GEOMETRIC STIFFNESS STRESS STIFFNESS
The component of the stiffness matrix that arises from the rotation of the internal stresses in a large deflection problem. This stiffness is positive for tensile stresses and negative for compressive stresses. If the compressive stresses are sufficiently high then the structure will buckle when the geometric stiffness cancels the elastic stiffness.
Errors in the geometrical representation of the model. These generally arise from the approximations inherent in the finite element approximation.
GLOBAL STIFFNESS MATRIX
The assembled stiffness matrix of the complete structure.
Deformations sufficiently high to make it necessary to include their effect in the solution process. The problem requires a large deflection non-linear analysis.
The subspace iteration (simultaneous vector iteration) method uses extra guard vectors in addition to the number of vectors requested by the user. These guard the desired vectors from being contaminated by the higher mode vectors and speed up convergence.
GUYAN REDUCTION METHOD
A method for reducing the number of degrees of freedom in a dynamic analysis. It is based upon a static approximation and always introduces some error in the computed dynamic solution. The error depends upon the choice of master freedoms.
Forces arising from Coriolis acceleration. These can destabilize a dynamic response and cause whirling.
A structure where the stiffness increases with load.
A dynamic loading that is periodic and can be represented by a Fourier series.
The analysis of the steady state heat flow within solids and fluids. The equilibrium balance between internal and external heat flows.
HERMITIAN SHAPE FUNCTIONS
Shape functions that provide both variable and variable first derivative continuity (displacement and slope continuity in structural terms) across element boundaries.
Type of 3D element which has six quadrilateral faces.
HIDDEN LINE REMOVAL
Graphical plots of models where non-visible mesh lines are not plotted.
HIGH ASPECT RATIO LOW ASPECT RATIO
The ratio of the longest side length of a body to the shortest is termed its aspect ratio. Generally bodies with high aspect ratios (long and thin) are more ill conditioned for numerical solution than bodies with an aspect ratio of one.
Constraints that can be defined for any magnitude of displacement.
The material property equations relating stress to strain for linear elasticity. They involve the material properties of Young’s modulus and Poisson ratio.
Zero energy modes of low order quadrilateral and brick elements that arise from using reduced integration. These modes can propagate through the complete body.
Convergence towards a more accurate solution by subdividing the elements into a number of smaller elements. This approach is referred to as H – convergence because of improved discretization due to reduced element size.
A finite element method which requires an increasing number of elements to improve the solution.
Making the mesh finer over parts or all of the body is termed h -refinement. Making the element order higher is termed p -refinement.
Elements that use stress interpolation within their volume and displacement interpolation around their boundary.
The stress arising from a uniform pressure load on a cube of material. It is the average value of the direct stress component s at any point in the body.
A damping model representing internal material loss damping. The energy loss per unit cycle is independent of frequency. It is only valid for harmonic response.
Numerical (rounding) errors that arise when using ill- conditioned equations.
ILL-CONDITIONING ILL-CONDITIONED EQUATIONS
Equations that are sensitive to rounding errors in a numerical operation. The numerical operation must also be defined. Equations can be ill conditioned for solving simultaneous equations but not for finding eigenvalues.
IMPULSE RESPONSE FUNCTION
The response of the system to an applied impulse.
IMPULSE RESPONSE MATRIX
The matrix of all system responses to all possible impulses. It is always symmetric for linear systems. It is the inverse Fourier transform of the dynamic flexibility matrix.
A solutions process that involves applying the loading in small increments and finding the equilibrium conditions at the end of each step. Such solutions a re generally used for solving non-linear problems.
INELASTIC MATERIAL BEHAVIOR
A material behavior where residual stresses or strains can remain in the body after a loading cycle, typically plasticity and creep.
The ratio of the steady state acceleration response to the value of the forcing function for a sinusoidal excitation.
The force that is equal to the mass times the acceleration.
The load at which a structure first buckles.
The components of the strains that are non-elastic. Typically thermal strain and plastic strain.
INTEGRATION BY PARTS
A method of integrating a function where high order derivative terms are partially integrated to reduce their order.
INTERPOLATION FUNCTIONS SHAPE FUNCTIONS
The polynomial functions used to define the form of interpolation within an element. When these are expressed as interpolations associated with each node they become the element shape functions.
Elements that use the same shape functions (interpolations) to define the geometry as were used to define the displacements. If these elements satisfy the convergence requirements of constant stress representation and strain free rigid body motions for one geometry then it will satisfy the conditions for any geometry.
Materials where the material properties are independent of the co -ordinate system.
A method for finding eigenvalues and eigenvectors of a symmetric matrix.
A square matrix relating derivatives of a variable in one coordinate system to the derivatives of the same variable in a second coordinate system. It arises when the chain rule for differentiation is written in matrix form.
A method for finding the stress intensity factor for fracture mechanics problems.
The interconnections between components. Joints can be difficult to model in finite element terms but they can significantly affect dynamic behavior.
KINEMATIC BOUNDARY CONDITIONS
The necessary displacement boundary conditions for a structural analysis. These are the essential boundary conditions in a finite element analysis.
KINEMATICALLY EQUIVALENT FORCES (LOADS)
A method for finding equivalent nodal loads when the actual load is distributed over a surface of a volume. The element shape functions are used so that the virtual work done by the equivalent loads is equal to the virtual work done by the real loads over the same virtual displacements. This gives the most accurate load representation for the finite element model. These are the non-essential stress boundary conditions in a finite element analysis.
KINEMATICALLY EQUIVALENT MASS
If the mass and stiffness are defined by the same displacement assumptions then a kinematically equivalent mass matrix is produced. This is not a diagonal (lumped) mass matrix.
The energy stored in the system arising from its velocity. In some cases it can also be a function of the structural displacements.