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NAFEMS Glossary of Terms J-Q

JACOBI ITERATION METHODAn iterative method for the solution of a system of simultaneous linear algebraic equations in which the dependent variable at each grid point is solved using initial guess values for the neighbouring points of previously computed values.
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 (q.v.).
JAMESON’S MULTISTAGE METHODRunge-Kutta type method that is used for the solution of Euler’s equation.
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.
JURY PROBLEMSProblems involving elliptic   equations where the solution within the domain depends on the total boundary around the domain.
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 (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 LOADSThe 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 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.
KOLMOGOROFF SCALEIn a turbulent flow, the scale associated with the smallest eddies.
KUTTA CONDITIONThis requires that equal static pressures exist on both sides of the sharp trailing edge of an airfoil. It is required in potential  flow calculations to obtain a solution to lifting airfoils. It is also imposed in some specialist numerical solvers of the Euler equation.
K- TURBULENCE MODELA two-equation  turbulence  model, formulated by the use of the eddy-viscosity hypothesis, where the effect of turbulence is captured by the fluid turbulent kinetic energy (k) and energy dissipation rate ().
LAGRANGE INTERPOLATION 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.
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 (q.v. also Eulerian formulation).
LAGRANGIAN FRAME OF REFERENCEA frame of reference that moves with a particle or element of fluid. The equations of fluid flow can be derived in this frame of reference. Methods for solving the dynamics of particles or fluids by ‘tracking’ their position in space relative to a fixed reference frame are referred to as Lagrangian methods.
LAMINAR FLAMELET MODELA reaction model that considers a turbulent flame front to be represented as an array of laminar ‘flamelets’.
LAMINAR FLOWFlow in which fluid moves in layers, without turbulence. Diffusive and dissipative effects take place only by molecular diffusion. Laminar flow usually exists at low Reynolds  numbers. In practice, for normally encountered flows of low viscosity (air, water), laminar flow only occurs at low velocity or very small physical length scales. However boundary  layers can exhibit laminar behaviour even at high speeds and on relatively large engineering components, because the dominant length scale is the small thickness of the boundary layer.
LAMINAR SUB-LAYERIn a turbulent boundary layer, the region of fluid closest to the wall where the fluid motion is dominated by viscous effects and turbulent motion is suppressed by viscous action.
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.
LAPLACE EQUATIONA steady-state transport equation for a variable  of the form:
LARGE DISPLACEMENTSDisplacements that are sufficiently large to render small displacement theory invalid.
LARGE EDDY SIMULATION (LES) TURBULENCE MODELLINGThis may be considered a compromise between direct numerical simulation (DNS) and the use of turbulence models (RANS). The unsteady flow equations are solved for the mean flow and larger eddies and a ‘sub-grid scale’ model is used to simulate the effects of the smaller eddies. Since it is the largest eddies which contain the most energy and interact most strongly with the mean flow, the LES approach results in a good model of the main effects of the turbulence. Since the grid size no longer has to be small enough to allow for the smallest turbulent eddies, this method is much less computationally expensive than DNS and may be applied to a wider range of flows. However, time dependant simulations using relatively fine meshes are still necessary, so the computational requirement is still high.
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.
LAW OF THE WALLAn assumed log law profile adjacent to the wall, which models the effect of the wall across the boundary layer.
LAX-WENDROFF METHODAn explicit finite  difference method particularly suited to marching numerical solutions, either in space or time. Similar to MacCormack’s method.
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.
LEAPFROG METHODA three level discretisation  scheme for unsteady  flows based on the mid-point integration rule.
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.
LEITH’S SCHEMEA central difference based version of the QUICKEST discretisation scheme (which was developed prior to QUICKEST).
LES TURBULENCE MODELLINGSee large eddy simulation turbulence modelling.
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 RELAXATIONAn iterative method in which variables are solved on a series of lines perpendicular to the flow and sweeping through it.
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.
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.
LINEARISATIONApproximation of local variations of a parameter by a linear form.
LOAD CONTROLA means of advancing a non-linear solution using a load parameter: this is the conventional method, others being displacement and arc length control.
LOADINGIn the finite element method, the definition of field quantities that impart energy to the structure and are therefore the reason for the analysis.
LOADINGSThe loads applied to a structure that result in deflections and consequent strains and stresses.
LOCAL STRESSESAreas of stress that are significantly different from (usually higher than) the general stress level.
LOCAL TIME STEPPINGA convergence acceleration technique used for steady  state problems which are solved using a time dependant method, in which local time steps vary between cells.
LOCALISATIONFor softening materials, a tendency for non-linear behaviour to concentrate into local bands, requiring special treatment.
LOGARITHMIC STRAINSee true strain.
LOG-LAWA logarithmic profile assumed to represent the positional variation of a variable close to the wall in a boundary layer. See law of the wall.
LOW REYNOLDS NUMBER TURBULENCE MODELA turbulence model that is valid for low Reynolds Number flow regimes, usually close to boundaries.
LOWER BOUND SOLUTION UPPER 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. 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.
LU FACTORISATIONDecomposition of a matrix into ‘upper triangular’ and ‘lower triangular’ forms. This can lead to an easy solution of the resulting triangular set of equations.
LUMPED MASS MODELThe system mass is represented by a number of point masses or particles. The mass matrix is diagonal.
MACCORMACK’S METHODAn explicit finite  difference  method particularly suited to marching numerical solutions, either in space or time. Similar to the Lax-Wendroff method.
MACH NUMBERA dimensionless number that is the ratio of the speed of fluid flow to the speed of sound in the fluid.
MAGNUSSON MODELA commonly used reaction model in which the rate of a reaction is calculated based on the finite chemistry and also based on the turbulence (generally using the eddy break-up model). The slower of the two rates is assumed to govern the reaction process and is used to obtain the solution.
MARANGONI CONVECTIONSee capillary convection.
MARANGONI NUMBERA number characterising the thermal forces at the surface of a free surface flow.
MARCHINGAn explicit method that computes a solution at a given step as a function of known values at the previous step.
MARKER-AND-CELL METHODA type of free surface method in which weightless markers in each cell are used to track the free surface profile. It is similar to the volume of fluid method.
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 or substructuring methods.
MATERIAL DATAThe data required to specify to the finite element process the relevant material properties.
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 MATRIX MATERIAL FLEXIBILITY MATRIXThe 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 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.
MATRIX NOTATION MATRIX ALGEBRAA 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 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.
MESHSee grid.
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-convergence) or increase in order of element type (p-convergence) to produce improvements in solution accuracy.
MESH DENSITY MESH REFINEMENTThe 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 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 GENERATION ELEMENT GENERATIONThe process of generating a mesh of elements over the structure. This is normally done automatically or semi-automatically.
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.
METHOD OF CHARACTERISTICSA method for solving a set of hyperbolic equations by setting out the equations in the form of ‘characteristics’ that relate the variables uniquely.
METHOD OF LINESMethod by which the discretised transport equations can be expressed as temporal ordinary differential equations. In this way, any standard ordinary differential equation approach can be used in the solution process.
MINDLIN ELEMENTSA form of thick shell element.
MIXED HARDENINGA combination of isotropic and kinematic hardening.
MIXING LENGTHIn turbulent flow, a characteristic (or typical) distance travelled by fluid particles in a direction normal to the flow (also known as Prandtl’s  mixing  length) or a characteristic size for turbulent eddies.
MIXTURE FRACTIONA non-dimensional variable used to describe the relative quantities of two species or phases.
MOBILITYThe ratio of the steady state velocity response to the value of the forcing function for a sinusoidal excitation.
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.
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 eigenvector contains an arbitrary normalising factor but the ratio of modal stiffness to modal mass is always the eigenvalue.
MODAL TESTINGThe experimental technique for measuring resonant frequencies (eigenvalues) and mode shapes (eigenvectors).
MODE PARTICIPATION FACTORThe generalised force in each modal equation of a dynamic system.
MODE SHAPESame as eigenvector (q.v.).
MODEL ACCURACYA measure of the similarity of a conceptual model to the physical flow it is intended to represent and one of the measures by which a solution is validated.
MODELLINGThe process of idealising a system and its loading to produce a numerical (finite element) model.
MODELLING ERRORSErrors due to the difference between the physical flow and the exact solution of the mathematical models being solved.
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.
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.
MOMENTUM EQUATIONAn expression of Newton’s Second Law, where the rate of change of momentum equals the sum of forces on a fluid particle.
MOONEY-RIVLIN STRAIN ENERGY FUNCTIONIs used in large strain elasticity (hyperelastic) problems, and is expressed in terms of the three strain invariants.
MOVING MESHA mesh which is updated with time, in order for moving boundary problems to be analysed.
MULTI DEGREE OF FREEDOMThe system is defined by more than one force/displacement equation.
MULTI-BLOCK APPROACHA meshing technique in which several 'blocks' of structured grid are combined to enable the meshing of complex geometries.
MULTIGRID METHODA technique used to accelerate the convergence of iterative  solution techniques based on the solution of a set of simultaneous correction equations. Adding neighbouring discretised equation coefficients generates these correction equations. The coefficient addition enables the correction equations to be solved for a smaller (and hence faster to solve) array than that used for the main discretisation equations. Also known as geometric multigrid, it is an alternative to algebraic multigrid.
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.
MULTIPHASE FLOWFlow consisting of two or more phases (gas, liquid, solid), e.g. gas bubbles rising through a liquid.
MULTI-POINT CONSTRAINTSWhere the constraint is defined by a relationship between more than one displacement at different node points.
MULTISTAGE METHODSee Runge-Kutta time stepping.
MUSCL APPROACHMonotone Upstream-centred Schemes for Conservation Laws. A method for the generation of second order upwind schemes via variable extrapolation.
NAHME NUMBERSee Griffith number.
NATURAL FREQUENCY (ALSO CALLED RESONANT FREQUENCY)The frequency at which resonance occurs, that is when the stiffness and the inertia forces cancel.
NATURAL MODESame as eigenvector (q.v.).
NATURAL STRAINSee true strain.
NAVIER-STOKES EQUATIONSPartial differential equations defining the unsteady viscous flow of fluids.
NEIGHBOUR COEFFICIENTSCoefficients used to simplify discretised governing equations.
NEIGHBOURING NODESThe nodes adjacent to the node of a control volume in the Finite Volume method.
NEUMANN BOUNDARY CONDITIONWhere values of flow variable derivatives are imposed on the boundaries of the flow domain.
NEWMARK METHOD NEWMARK BETA METHODAn implicit solution method for integrating second order equations of motion. It can be made unconditionally stable.
NEWMARK’S TIME STEPPING SCHEMESA family of time integration methods for the solution of transient dynamic problems.
NEWTON COTES FORMULAEA family of methods for numerically integrating a function.
NEWTON’S METHODA method for solving non-linear equations using inner and linearised outer iterations.
NEWTONIAN FLUIDFluid in which viscous shear stresses are assumed proportional to the velocity gradient perpendicular to the flow direction and the constant of proportionality is independent of the flow field.
NEWTON-RAPHSON METHODAn incremental-iterative non-linear procedure to solve the equilibrium equations: the tangential stiffness matrix is updated during every iteration of every increment.
NEWTON-RAPHSON NON-LINEAR SOLUTIONA general technique for solving non-linear equations. If the function and its derivative are known at any point then the Newton-Raphson method is second order convergent.
NO SLIP CONDITIONWhere velocity components at a solid wall are set equal to the velocity of the walls, i.e. the fluid does not slip over the wall but exhibits a velocity gradient from stationary flow at the wall to the free stream velocity.
NODAL POINTSee grid point.
NODAL VALUESThe value of variables at the node points. For a structure typical possible nodal values are force, displacement, temperature, velocity, x, y, and z.
NODESee grid point.
NODESThe element behaviour is defined by the response at the nodes of the elements. Nodes are always at the corners of the element, higher order elements have nodes at mid-edge or other edge positions and some elements have nodes on faces or within the element volume. The behaviour of the element is defined by the variables at the node. For a stiffness matrix the variables are the structural displacement, For a heat conduction analysis the nodal variable is the temperature. Other problems have other nodal variables.
NON-ASSOCIATIVE PLASTICITYA form of plasticity in which the yield function is not identical to the plastic potential.
NON-CONFORMING ELEMENTSElements that do not satisfy compatibility either within the element or across element boundaries or both. Such elements are not generally reliable although they might give very good solutions in some circumstances.
NON-CONSERVATION FORM OF EQUATIONSPartial differential equations obtained for an infinitesimal fluid element that travels through the flow (as opposed to being fixed in space and which therefore results in the conservation form of the equations).
NON-DIMENSIONAL FORM OF EQUATIONSAn equation form in which each term is non-dimensionalised using reference values for length, time, velocity, pressure etc.. This process potentially lends itself to a better appreciation of the dominant flow physics.
NON-HOLONOMIC CONSTRAINTSConstraints that can only be defined at the level of infinitesimal displacements. They cannot be integrated to give global constraints.
NON-LINEAR SYSTEM NON-LINEAR ANALYSISWhen at least one of the coefficients of stiffness, mass or damping vary with displacement or time then the system is non-linear. Superposition cannot be used to solve the problem.
NON-ORTHOGONAL GRID SYSTEMA system in which the grid lines are not always at right angles, enabling meshing around irregular geometries.
NON-REFLECTING BOUNDARY CONDITIONA boundary condition that does not permit the reflection of pressure waves.
NON-STATIONARY RANDOMA force or response that is random and its statistical properties vary with time.
NON-STRUCTURAL MASSMass that is present in the system and will affect the dynamic response but it is not a part of the structural mass (e.g. the payload).
NON-UNIFORM GRIDSA grid in which the computational cells vary in size and / or shape.
NORMA scalar measure of the magnitude of a vector or a matrix.
NORMALISATIONThe adjustment of a series of values, using division by a constant, to provide a consistent reference value.
NORMALITY RULEA particular plastic flow rule to ensure that the plastic strain components are in a ratio such that their resultant is in a direction normal to the yield surface (q.v. also flow rule).
NORTON LAW/NORTON-BAILEY LAWA law for steady state creep with strain rate proportional to a power of stress.
NUMERICAL ACCURACYA measure of the accuracy of the numerical treatment (i.e. discretisation and convergence) and one of the measures by which a solution is verified.
NUMERICAL DIFFUSIONA type of numerical error that smears simulated flow gradients giving the same effect as flow diffusion. It is due to truncation  errors that arise as a result of representing the fluid flow equations in discrete form. It is inversely related to the grid resolution. Numerical diffusion may also be reduced by the use of higher order discretisation schemes and alignment of the grid lines with the streamlines. It is also known as false diffusion and it results in a diffusive error.
NUMERICAL DISPERSIONA numerical effect on the solution in Fourier space in which waves are spread in space, but not changed in amplitude.
NUMERICAL DISSIPATIONA numerical effect on the solution in Fourier space in which the variation of the coefficients (or amplitude) is reduced.
NUMERICAL GRIDSee grid.
NUMERICAL INSTABILITYAn increasing oscillation of an iterative  solution or the growth of errors due to round-off or truncation in a numerical scheme.
NUMERICAL INTEGRATIONThe process of integrating the element stiffness matrix based on numerical algorithms such as Gaussian quadrature. Evaluations are made at strategic points within each element, known as Gauss points (q.v.).
NUMERICAL VISCOSITYAn error resulting from finite difference approximations causing excess energy dissipation.
O-GRIDA curvilinear grid wrapped around a smooth body to form the shape of an O.
ONE-SIDED DIFFERENCINGFirst order numerical differentiation (forward or backward) as opposed to second order central differencing.
ONE-STEP REACTIONSimplest chemical reaction. One group of reactants forms one group of products directly.
OPERATOR SPLITTINGDecomposing a differential equation solution scheme into several stages.
OPTIMAL POINTSStrategic locations within elements where stress evaluations are especially accurate, often at the Gauss point (q.v.) locations.
OPTIMAL SAMPLING POINTSThe minimum number of Gauss points required to integrate an element matrix. Also the Gauss points at which the stresses are most accurate (see reduced Gauss points).
ORDER OF ACCURACYThe number of terms retained in the series expansion used to approximate the equations in their discretised form.
ORTHOGONAL GRIDA grid in which intersections of grid lines are all perpendicular or near perpendicular.
ORTHONORMAL FUNCTIONSA pair of functions that are orthogonal to each other and normalised.
ORTHOTROPYA material where the response to load depends on the direction within the material. It is less general than anisotropy, and up to 12 independent constants are required to relate stress and strain.
OSCILLATORY PRESSURE SOLUTIONSSee checker-board pressure field.
OUTER ITERATIONSProgress of differential equation solutions from one (false) time step to the next after resolving non-linearity of the difference equations.
OUTFLOW BOUNDARY CONDITIONBoundary with specified flow rate out of the flow domain.
OUTLET BOUNDARY CONDITIONBoundary at which fluid leaves the flow domain. Often specified as a constant pressure boundary.
OVER DAMPED SYSTEMA system that has an equation of motion where the damping is greater than critical. It has an exponentially decaying, non-oscillatory impulse response.
OVER-RELAXATIONExtrapolation of results from one iteration to the next, often leading to instability.
OVERSTIFF SOLUTIONSLower bound solutions. These are associated with the assumed displacement method.
PANEL METHODSA rapid computational method for determining the value of a potential function (and hence surface velocities and pressures) on discrete panels representing a surface geometry. The method is suited to flows that can be considered inviscid, incompressible and irrotational, features typical of many high speed vehicle aerodynamic applications. Boundary layer calculations, compressibility corrections and wake surfaces extend the use and accuracy of panel method calculations.
PARABOLIC EQUATIONPartial differential equations of the form Auxx + 2Buxy + Cuyy = F(x,y,u,ux,uy) for which AC - B2 = 0. Examples of problems that are described by parabolic equations include many time dependant problems such as unsteady viscous flow and unsteady heat conduction. Steady problems can also be parabolic; for example, equations describing one-dimensional supersonic flow fall into this category. In a physical sense, parabolic equations describe behaviour in which the influence of a perturbation extends in only one direction in either time or space. Examples are developing viscous flow in a duct or pipe, or attached (unseparated) boundary layer flow. Here the equations are such that the dominant terms governing transport in the streamwise direction are those describing convection. As long as the flow remains attached with no local flow reversal, these convective terms always operate in the bulk flow downstream direction.
PARAMETRIC STUDIES PILOT STUDIESInitial studies conducted on small simplified models to determine the important parameters in the solution of a problem. These are often used to determine the basic mesh density required.
PARTICIPATION FACTORThe fraction of the mass that is active for a given mode with a given distribution of dynamic loads. Often this is only defined for a specific load case of inertia (seismic) loads.
PARTICLE SOURCE IN CELL (PSIC) METHODA multiphase Eulerian-Lagrangian method in which a low volume dispersed phase is modelled as discrete sources of mass, momentum and energy in the continuum flow field.
PARTICLE TRAJECTORIESPaths of discrete material elements suspended in a continuum fluid.
PATCH TESTA simple element type test using a patch of several elements, one of which is arbitrarily orientated with respect to the global co-ordinates. If the patch is loaded by displacements consistent with a state of constant strain and the strain inside the selected element is constant, the test is passed.
PATCHINGThe pre-iterative definition of flow variables at specific locations.
PATHLINEAn imaginary line which represents the path travelled by an individual particle of fluid. In steady flow, pathlines coincide with streamlines.
PDF (PROBABILITY DENSITY FUNCTION)Description of the probability of an event at a given value of an independent variable, especially used in reactions.
PECLET NUMBERDimensionless number that is the ratio of convection to diffusion (mass transfer) or conduction (heat transfer). It is the equivalent of the local or cell Reynolds number.
PENALTY FORMULATIONA means of reducing the computational effort in incompressible flow problems by eliminating the continuity equation as well as the pressure term from the momentum equations. Pressure can subsequently be recovered from the computed velocity field.
PENALTY FUNCTION (ALSO CALLED PENALTY STIFFNESS)In the context of contact algorithms, a constraint on stiffness behaviour usually applied via large numbers in the equations, e.g. by introducing stiff springs.
PENTADIAGONAL MATRIXA matrix containing zeroes in all elements except the diagonal, the subdiagonal, the superdiagonal and slots adjacent (vertically or horizontally) to the subdiagonal and superdiagonal.
PERFECT PLASTICITYPlastic behaviour where the yield stress remains constant for all values of plastic strain.
PERIODIC BOUNDARY CONDITIONA boundary condition for flows which are periodic in space (see cyclic boundary condition) or periodic in time.
PERIODIC GRIDA grid used to represent geometries that repeat periodically in space. It is often used in conjunction with periodic boundary conditions.
PERIODIC RESPONSE (FORCE)A response (force) that regularly repeats itself exactly.
PETROV-GALERKINA finite  element  method of discretisation that uses modified weighting functions. The Petrov-Galerkin method differs from the Galerkin or Bubnow-Galerkin method in that the weighting functions are different from the interpolation functions.
PHASE ANGLEThe ratio of the in-phase component of a signal to its out-of-phase component gives the tangent of the phase angle of the signal relative to some reference.
PHASE CHANGEWhen a substance subjected to temperature changes transforms from solid to liquid or gaseous state. During this phase change, latent heat is either released or absorbed.
PHASE VELOCITYThe velocity of one component in a multi-phase flow.
PHYSICAL BOUNDARY CONDITIONA physical property at a boundary.
PHYSICAL PROPERTIESA description of the material characteristics (density, viscosity, thermal conductivity, heat capacity etc.).
PICARD ITERATIONThe solution of integral equations by successive iterations.
PISO ALGORITHMThe Pressure Implicit with Splitting Operators algorithm is a pressure velocity coupling algorithm involving one predictor and two corrector steps.
PLANE STRAIN PLANE STRESSA two dimensional analysis is plane stress if the stress in the third direction is assumed zero. This is valid if the dimension of the body in this direction is very small, e.g. a thin plate. A two dimensional analysis is plane strain if the strain in the third direction is assumed zero. This is valid if the dimension of the body in this direction is very large, e.g. a cross-sectional slice of a long body.
PLASTIC STRAINIrrecoverable permanent strain due to time independent plasticity.
PLASTIC ZONESRegions in a body where a stress measure (usually the equivalent stress) lies on the yield surface and plastic strains are accruing.
PLATE BENDING ELEMENTSTwo dimensional shell elements where the in plane behaviour of the element is ignored. Only the out of plane bending is considered.
PLY LAY-UPSee lay-up.
POINT SINKAn isolated point through which something (i.e. mass, momentum, energy) leaves a flow field.
POINT SOURCEAn isolated point from which something (i.e. mass, momentum, energy) issues into a flow field.
POINT-COLLOCATION METHODWeighting function for method of weighted residuals.
POISSON SOLVERSMethods of solving Poisson’s (elliptic) equations.
POISSONS RATIOThe material property in Hookes law relating strain in one direction arising from a stress in a perpendicular direction to this.
POLAR GRIDGrid based on spherical co-ordinates (two angles and radius).
POROUS JUMP BOUNDARY CONDITIONA one-dimensional version of porous media modelling used to model thin walled membranes, such as mesh screens, filter papers or perforated plates, which exhibit a known pressure drop. The pressure loss is simulated as occurring between adjacent cells.
POROUS MEDIA MODELLINGThe use of a geometric region operating as a momentum sink via terms for inertial and viscous resistance. This can be used to represent the pressure drop occurring through a variety of media including packed beds, tube banks, etc..
POSITIVE DEFINITE MATRIXA matrix is positive definite if the dot product with itself is equal to zero, if and only if, the matrix is itself zero.
POST ANALYSIS CHECKSChecks that can be made on the results after the analysis. For a stress analysis these could include how well stress free boundary conditions have been satisfied or how continuous stresses are across elements.
POST YIELD FRACTURE MECHANICS (PYFM, ALSO CALLED ELASTIC-PLASTIC FRACTURE MECHANICS, EPFM)A given crack inside a loaded structure behaves in conditions of PYFM when the crack fields local to the crack tip exhibit considerable plastic behaviour.
POST-PROCESSINGThe interrogation of the results after the analysis phase. This is usually done graphically.
POTENTIAL ENERGYThe energy associated with the static behaviour of a system. For a structure this is the strain energy.
POTENTIAL ENERGY RELEASE RATEFor a hypothetically small increase in crack length or area, this is the amount of potential energy released divided by that length or area. It equals the negative of the strain energy release rate (q.v.) when elastic conditions predominate. It provides the basis for fracture parameters in post yield fracture mechanics (q.v.) and other non-linear conditions.
POTENTIAL FLOWFluid flow problems where the flow can be represented by a scalar potential function.
POWER METHODA method for finding the lowest or the highest eigenvalue of a system.
PRANDTL NUMBERA dimensionless number that is the ratio of momentum diffusivity to thermal diffusivity.
PRANDTL’S MIXING LENGTHSee mixing length.
PRANDTL-REUSS EQUATIONSThe equations relating an increment of stress to an increment of plastic strain for a metal undergoing plastic flow.
PRANDTL-REUSS FLOW RULEIn plasticity theory, the special form of the normality rule corresponding to von Mises yield criterion (q.v.).
PRE-CONDITIONING (AND PRE-CONDITIONING MATRIX)A convergence acceleration technique used in iterative methods.
PREDICTOR STEPSee predictor-corrector method.
PREDICTOR-CORRECTOR METHODA method for integrating ordinary differential equations by extrapolating a polynomial fit of derivatives from previous points to new points (predictor step) This is then used to interpolate the derivative (corrector step).
PREDICTOR-CORRECTOR SCHEMESThe two-phase format of a time or load stepping scheme where the predicted solution is corrected prior to advancing to the next step.
PRE-PROCESSINGDefinition of the flow to be simulated (fluid properties, geometry, mesh generation, boundary conditions etc.).
PRESSURE BOUNDARY CONDITIONA boundary where the local pressure is defined.
PRESSURE COEFFICIENTA dimensionless description of local pressure that has several different definitions according to the application.
PRESSURE CORRECTIONA modified form of the continuity equation that is used with momentum equations to solve for pressure and velocities (e.g. SIMPLE algorithm).
PRESSURE–VELOCITY COUPLINGThe linkage of pressure and velocity in the simultaneous solution of momentum and continuity equations.
PRESTO SCHEMEThe PREssure STaggering Option is a method of calculating the cell face pressure using a continuity balance for a ‘staggered’ cell, centred on the cell face.
PRIMARY COMPONENTThose parts of the structure that are of direct interest for the analysis. Other parts are secondary components.
PRIMARY CREEPThe initial part of a creep test where the strain rate is decreasing.
PRIMITIVE VARIABLESDensity, pressure and velocity components.
PRINCIPAL CURVATUREThe maximum and minimum radii of curvature at a point.
PRINCIPAL PLANESThe planes on which the shear stresses are zero. Three such planes exist at every point in a stressed body.
PRINCIPAL STRESSESStresses normal to the principal planes.
PROBABILITY DENSITY FUNCTIONSee PDF.
PROFILEThe profile of a symmetric matrix is the sum of the number of terms in the lower (or upper) triangle of the matrix ignoring the leading zeros in each row. Embedded zeros are included in the count. It gives a measure of the work required to factorise the matrix when using the Cholesky solution. It is minimised by node renumbering.
PROJECTION METHODA method in which a velocity field is constructed then corrected to satisfy continuity.
PROLONGATIONA method to interpolate a value/correction from coarse to fine grids (multi-grid).
PROPORTIONAL DAMPINGA damping matrix that is a linear combination of the mass and stiffness matrices. The eigenvectors of a proportionally damped system are identical to those of the undamped system.
PROPORTIONAL LOADINGOccurs when all the external loads are applied simultaneously, and increase in proportion to one another throughout the loading history. This clearly does not occur when one component of load is applied and then another.
PSEUDO-COMPRESSIBILITYA method of adding an artificial compressibility term to the continuity equation to solve the incompressible Navier-Stokes    equations using time dependent compressible methods.
PSEUDO-PATH LINEA vector contained within a characteristic surface.
PSEUDO-TRANSIENT METHODSee pseudo-unsteady formulation.
PSEUDO-UNSTEADY FORMULATIONA time-marching technique following the numerical solution in time until steady- state is reached.
PSEUDO-VELOCITYA type of guessed velocity used in the SIMPLER algorithm.
PSIC METHODSee particle source in cell method.
QR METHODA technique for finding eigenvalues. This is currently the most stable method for finding eigenvalues but it is restricted in the size of problem that it can solve.
QUADRATIC INTERPOLATIONSee QUICK and QUICKEST.
QUADRATIC UPWIND DIFFERENCING SCHEMESee QUICK and QUICKEST.
QUADRIDIAGONAL MATRIXMatrix with four diagonally arranged terms.
QUADRILATERAL ELEMENTSFour sided, two dimensional elements.
QUASI-LINEAR DIFFERENTIAL EQUATIONSNon-linear equations that are assumed to contain locally-constant coefficients.
QUASI-ONE-DIMENSIONAL NOZZLE FLOWSFlows for which the flow properties are assumed to vary in the axial direction only.
QUICK UPWIND SCHEMEA third-order accurate upstream-weighted quadratic interpolation scheme. The QUICK scheme, as its name implies, uses a quadratic function that passes through three node values, to determine the required node value. It computes the cell boundary value of the variable based on the values in the two adjacent cell centres and at a third cell centre at an additional upstream point. See discretisation scheme.
QUICKEST UPWIND SCHEMEThe QUICK  scheme estimated, developed for the unsteady advection-diffusion equation and only used with explicit solvers.