Glossary Terms J-M

The NAFEMS Glossary


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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.
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 (q.v.).
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.
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 Loads
The point loads that are applied at the nodes of an element to represent a distributed load and that have been derived analytically to give the same work done as the distributed load.
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 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.
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 (q.v. 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.
Laminar Flow
A state of fluid motion where the fluid moves in layers without turbulence (q.v.).
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 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.
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 arc length control.
In the finite element method, the definition of field quantities that impart energy to the structure and are therefore the reason for the analysis.
The loads applied to a structure that result in deflections and consequent strains and stresses.
Local Stresses
Areas of stress that are significantly different from (usually higher than) the general stress level.
For softening materials, a tendency for non-linear behaviour to concentrate into local bands, requiring special treatment.
Logarithmic Strain
See true strain.
Lower Bound Solution Upper 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. The assumed stress form of the finite element solution gives an upper bound on the maximum stresses and strain energy (i.e. these are over estimated) for a given set of displacements.
Lumped Mass Model
The system mass is represented by a number of point masses or particles. The mass matrix is diagonal.
Mach Number
The ratio of the speed of a flowing fluid to the speed of sound in the fluid.
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 or substructuring methods.
Material Data
The data required to specify to the finite element process the relevant material properties.
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 Material Flexibility Matrix
The material stiffness matrix allows the stresses to be found from a given set of strains at a point. The material flexibility is the inverse of this, allowing the strains to be found from a given set of stresses. Both of these matrices must be symmetric and positive definite.
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.
Matrix Notation Matrix Algebra
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. Matrix algebra can be easily programmed on a digital computer.
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 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-convergence) or increase in order of element type (p-convergence) to produce improvements in solution accuracy.
Mesh Density Mesh Refinement
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. Making the mesh finer is generally referred to as h-refinement. Making the element order higher is referred to as p-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 Element Generation
The process of generating a mesh of elements over the structure. This is normally done automatically or semi-automatically.
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 and kinematic hardening.
The ratio of the steady state velocity response to the value of the forcing function for a sinusoidal excitation.
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.
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 eigenvector contains an arbitrary normalising factor but the ratio of modal stiffness to modal mass is always the eigenvalue.
Modal Testing
The experimental technique for measuring resonant frequencies (eigenvalues) and mode shapes (eigenvectors).
Mode Participation Factor
The generalised force in each modal equation of a dynamic system.
Mode Shape
Same as eigenvector (q.v.).
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.
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.
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.