# The NAFEMS Glossary

# Terms J-M

Jacobi MethodA method for finding Eigenvalues and eigenvectors of a symmetric matrix.

Jacobian MatrixA 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.

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

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

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

Kinematic HardeningThis 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 / LoadsA 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 MassIf 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 EnergyThe energy stored in a system arising from its velocity. In some cases, it can also be a function of the structural displacements.

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

Lagrange MultipliersCoefficients arising from extra stiffness equations that represent additional constraint equations.

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

Lagrangian MethodFor 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 FunctionsA 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 FlowA state of fluid motion where the fluid moves in layers without Turbulence.

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

Lanczos MethodA 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 DisplacementsDisplacements that are sufficiently large to render small displacement theory invalid.

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

Large StrainsStrains that are sufficiently large to render small strain theory invalid.

Latent RootsSee Eigenvalues.

Latent VectorsSee Eigenvectors.

Lay-UpLay 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 FitMinimisation 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 LoadThe maximum load a structure can sustain without causing the structure to collapse.

Limit PointsPoints 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 SearchA technique for accelerating incremental-iterative solution procedures.

Line Spring AnalysisA 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 DependenceOne 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 SystemWhen 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 ControlA means of advancing a non-linear solution using a load parameter: this is the conventional method, others being displacement and theArc Length Method.

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

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

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

Logarithmic StrainSeeTrue Strain.

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

Mach NumberThe ratio of the speed of a flowing fluid to the speed of sound in the fluid.

MassThe 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 MatrixThe 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 FreedomsThe freedoms chosen to control the structural response when using a Guyan Reduction Method or substructuring methods.

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

Material Flexibility MatrixThe 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 FactorA measure of the damping inherent within a material when it is dynamically loaded.

Material PropertiesThe 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 MatrixThe 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 AlgebraMatrix algebra can be easily programmed on a digital computer. See also Matrix Notation.

Matrix Displacement MethodA form (the standard form) of the finite element method where displacements are assumed over the element. This gives a Lower Bound Solution.

Matrix Force MethodA form of the finite element method where stresses (internal forces) are assumed over the element. This gives an Upper Bound Solution.

Matrix InverseIf 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 NotationA 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 ProductsTwo 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 TransposeThe process of interchanging rows and columns of a matrix so that the j'th column becomes the j'th row.

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

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

Mesh ConvergenceThe 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 DensityThe 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 DesignThe 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 GenerationThe 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 RefinementThere 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 SpecificationThe process of choosing and specifying a suitable mesh of elements for an analysis.

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

Mindlin ElementsA form of thick shell element.

Mixed HardeningA combination of Isotropic Hardening and kinematic hardening.

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

Modal CondensationSee Condensation.

Modal DampingThe 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 MassThe 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 StiffnessThe 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 TestingThe experimental technique for measuring resonant frequencies (eigenvalues) and mode shapes (eigenvectors).Eigenvalues and mode shapes (eigenvectors).

Mode Participation FactorThe generalised force in each modal equation of a dynamic system.

Mode ShapeSee Eigenvectors.

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

Modes Of FractureThree 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 MethodA 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 StressA form of equivalent stress that includes the effects of friction in granular (e.g. sand) materials.

Mohr-Coulomb FrictionFrictional behaviour between surfaces in contact when relative slippage is governed by the coefficient of friction.

Mohr-Coulomb Yield CriterionA 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 FunctionIs used in large strain elasticity (hyperelastic) problems, and is expressed in terms of the three strain invariants. See also Hyperelasticity.

Multi Degree Of FreedomThe system is defined by more than one force/displacement equation.

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