The NAFEMS Glossary

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Terms J-M of the glossary can be found below
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Terms J-M

Jacobi Method

A method for finding Eigenvalues and eigenvectors of a symmetric matrix.

Jacobian 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.

Jacobians

A mathematical quantity reflecting the distortion of an element from the theoretically perfect shape for that element type. It can be used as an Element Shape Parameter.

J-Integral Method

Used in fracture mechanics to evaluate fracture parameters at a single crack tip calculated using an expression integrated along a path surrounding the tip.

Joints

The interconnections between components. Joints can be difficult to model in finite element terms but they can significantly effect dynamic behaviour.

Kinematic Boundary Conditions

The necessary displacement boundary conditions for a structural analysis. These are the essential boundary conditions in a finite element analysis.

Kinematic Hardening

This occurs when, as plastic strains increase after initial yielding, the yield surface in principal stress coordinates translates as a rigid body while maintaining its initial shape and orientation.

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 kinetically equivalent mass matrix is produced. This is not a diagonal (lumped) mass matrix.

Kinetic Energy

The energy stored in a system arising from its velocity. In some cases, it can also be a function of the structural displacements.

Lagrange Interpolation

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.

Lagrange Multipliers

Coefficients arising from extra stiffness equations that represent additional constraint equations.

Lagrangian Formulation

A geometrically non-linear formulation where the equilibrium conditions are satisfied in the fixed reference configuration. See also Eulerian Formulation.

Lagrangian Method

For non-linear large deflection problems the equations can be defined in various ways. 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. See also Eulerian Method.

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.

Laminar Flow

A state of fluid motion where the fluid moves in layers without Turbulence.

Laminated Composite

A Composite Material where each fibre/resin layer is bonded to adjacent layers in the curing process.

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.

Large Displacements

Displacements that are sufficiently large to render small displacement theory invalid.

Large Rotations

Rotations that are sufficiently large to render small rotation theory invalid; relevant to beams, plates and shells.

Large Strains

Strains that are sufficiently large to render small strain theory invalid.

Latent Roots

See Eigenvalues.

Latent Vectors

See Eigenvectors.

Lay-Up

Lay up of individual plies or layers to form laminated material. Plies may be arranged in alternating fibre orientations to produce favourable multidirectional strength.

Least Squares Fit

Minimisation 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.

Limit Load

The maximum load a structure can sustain without causing the structure to collapse.

Limit Points

Points at which the tangent to the load-displacement curve becomes either horizontal or vertical and the structural stiffness matrix becomes singular under load or displacement control, respectively.

Line Search

A technique for accelerating incremental-iterative solution procedures.

Line Spring Analysis

A technique for modelling part-through cracks in shell type structures. An equivalent distributed spring replaces the crack with matching compliance, so the curvature is effectively ignored but the modelling is easier.

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. See also Singular Matrix.

Linear Elastic Fracture Mechanics (LEFM)

A given crack inside a loaded structure behaves in conditions of LEFM if the crack fields local to the crack tip are assumed to be elastic, and any plastic behaviour is neglected.

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.

Load Control

A means of advancing a non-linear solution using a load parameter: this is the conventional method, others being displacement and theArc Length Method.

Loading

In the finite element method, the definition of field quantities that impart energy to the structure and are therefore the reason for the analysis.

Local Stresses

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

Localisation

For softening materials, a tendency for non-linear behaviour to concentrate into local bands, requiring special treatment. See also Non-Linear Analysis.

Logarithmic Strain

SeeTrue Strain.

Lower 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. See also Strain Energy.

Lumped Mass Model

The system mass is represented by a number of point masses or particles. The mass matrix is diagonal. See also Diagonal Generalised Matrix.

Mach Number

The ratio of the speed of a flowing fluid to the speed of sound in the fluid.

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 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 Method or substructuring methods.

Material Data

The data required to specify to the finite element process the relevant material properties. See also Material Properties.

Material Flexibility Matrix

The material flexibility matrix is the inverse of the Material Stiffness Matrix allowing the strains to be found from a given set of stresses. The resulting matrix must be symmetric and positive definite.

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 behaviour 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

The material stiffness matrix allows the stresses to be found from a given set of strains at a point. This matrix must be symmetric and positive definite. See also Material Flexibility Matrix.

Matrix Algebra

Matrix algebra can be easily programmed on a digital computer. See also Matrix Notation.

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.See also Singular Matrix.

Matrix Notation

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. See also Matrix Algebra.

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 behaviour is where the strains are constant from the centre line of a beam or centre 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 centre line then this is called bending behaviour.

Mesh Adaptivity

The automatic alteration of meshes to provide refinement where the calculated variables vary rapidly and coarsening where they vary slowly.

Mesh Convergence

The progressive refinement of element size and positioning in mesh models (H-refinement) or increase in order of element type (p-refinement) to produce improvements in solution accuracy.

Mesh Density

The mesh density indicates the size of the elements in relation to the size of the body being analysed. 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. See also Mesh Refinement.

Mesh Design

The creation of a suitable mesh model, to represent the given structure with suitable refinement in regions of high field variation, good representation of boundaries, and incorporating all other required features.

Mesh Generation

The process of generating a mesh of elements over the structure. This is normally done automatically or semi-automatically. This is also referred to as Element Generation.

Mesh Refinement

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 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.

Mixed Hardening

A combination of Isotropic Hardening and kinematic hardening.

Mobility

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

Modal Condensation

See Condensation.

Modal Damping

The damping associated with the generalised displacements defined by the eigenvectors. Its value has no physical significance since the eigenvector contains an arbitrary normalising factor.eigenvectors contain an arbitrary normalising factors.

Modal Mass

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

Modal Stiffness

The stiffness associated with the generalised displacements defined by the eigenvectors. Its value has no physical significance since the eigenvectors contain arbitrary normalising factors but the ratio of modal stiffness to modal mass is always the Eigenvalues.

Modal Testing

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

Mode Participation Factor

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

Mode Shape

See Eigenvectors.

Modelling

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

Modes Of Fracture

Three separate deformation modes exist at any point along a crack profile, representing the basic effects of crack opening, shearing and tearing, commonly known as modes I, II and III. In practice, combinations of these modes are usually present.

Modified Newton-Raphson Method

A Newton-Raphson solution in which the tangent stiffness matrix is updated only at the beginning of every increment. See also Newton-Raphson Non-Linear Solution.

Mohr-Coulomb Equivalent Stress

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

Mohr-Coulomb Friction

Frictional behaviour between surfaces in contact when relative slippage is governed by the coefficient of friction.

Mohr-Coulomb Yield Criterion

A generalisation of the Coulomb friction failure law; used for concrete, rock and soils, where the hydrostatic stress does not influence yielding. The criterion is an inverted hexagonal pyramid in principal stress space.

Mooney-Rivlin Strain Energy Function

Is used in large strain elasticity (hyperelastic) problems, and is expressed in terms of the three strain invariants. See also Hyperelasticity.

Multi Degree Of Freedom

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

Multigrid Methods

Used in CFD to accelerate the convergence of iterative solution techniques based on the solution of a set of simultaneous correction equations, and allowing a reduction in the number of equations to be solved.

Multi-Point Constraints

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