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NAFEMS Glossary of Terms R-V

RADIATIONA mode of heat transfer due to electromagnetic waves. Thus, the heat energy can be transferred in a vacuum. It is characterised by the Stefan-Boltzmann law.
RADIATION DAMPINGDamping that arises from energy being carried away from a vibrating body by expanding pressure waves. Sound radiation is an example of this. Such radiating energy, both to the surrounding fluid and through the supports, often forms the main damping in a vibrating structure.
RAMBERG-OSGOOD POWER LAWA stress-strain relationship where the strain is proportional to a power of stress.
RANDOM ANALYSISWhen the applied loading is only known in terms of its statistical properties. The loading is non-deterministic in that its value is not known exactly at any time but its mean, mean square, variance and other statistical quantities are known.
RANDOM VIBRATIONSThe applied loading is only known in terms of its statistical properties. The loading is non-deterministic in that its value is not known exactly at any time but its mean, mean square, variance and other statistical quantities are known.
RANDOM VORTEX METHODAn algorithm capable of tracing the action of elementary turbulent eddies and their cumulative effects, without imposing any restriction upon their motion.
RANK DEFICIENCYA measure of how singular a matrix is.
RANKINE-HUGONIOT RELATIONSConservation equations across a steady, normal shock in terms of ratios.
RANS (REYNOLDS AVERAGED NAVIER STOKES)A form of the Navier  Stokes  equations in which additional terms (known as Reynolds  stresses) are included to account for the time averaged effects of turbulence. See Turbulence Models.
RATCHETTINGOccurs in cyclic loading (q.v.) when plastic strains keep on accumulating incrementally with each cycle, leading to eventual failure via incremental collapse.
RAYLEIGH DAMPINGA model for representing the variation of damping with frequency.
RAYLEIGH NUMBERDimensionless expression of the strength of a buoyant flow, with laminar/turbulent transition occurring typically between 108 and 1010.
RAYLELGH QUOTIENTThe ratio of stiffness times displacement squared (2*strain energy) to mass times displacement squared. The minimum values of the Rayleigh quotient are the eigenvalues.
REACTION FORCESThe forces generated at support points when a structure is loaded.
REALISABLESatisfying mathematical constraints due to physics.
RECEPTANCEThe ratio of the steady state displacement response to the value of the forcing function for a sinusoidal excitation. It is the same as the dynamic flexibility.
REDUCED INTEGRATIONThe process of intentionally under-integrating the element stiffness matrix to prevent problems such as shear locking or to improve the element’s performance.
REDUCED NAVIER-STOKES EQUATIONSThe Navier-Stokes  equations can be reduced for a variety of flow situations including incompressible flows (density variation terms removed) and isothermal flows (temperature variation terms removed).
REFERENCE PRESSUREFixed location absolute pressure value to which other pressures are related.
REGION OF DEPENDENCERegion within the Mach cone ahead of and able to affect a supersonic body.
REGION OF INFLUENCERegion within the Mach cone affected by passage of a supersonic body.
RELAXATION TECHNIQUEA finite  difference technique particularly suited for the solution of elliptic  partial differential equations.
RESIDUALError calculated from summing terms in partially converged equations.
RESIDUAL FORCESThe forces which are equal to the applied load minus the internal resisting forces which occur during non-linear solutions: used to measure the state of equilibrium by comparison to the convergence tolerance.
RESIDUAL NORMNormalised residual to help judge overall convergence.
RESONANT RESPONSEThe response of a system to vibratory forces.
RESPONSE SPECTRUM METHODA method for characterising a dynamic transient forcing function and the associated solution technique. It is used for seismic and shock type loads.
RESTARTS CHECKPOINTSThe process whereby an analysis can be stopped part way through and the analysis restarted at a later time.
RESTRICTIONMulti-grid term: smooth residual from fine to coarse grid.
REYNOLDS ANALOGYAnalogy between heat and momentum transfer where Prandtl Number is equal to one.
REYNOLDS NUMBERA dimensionless number that is the ratio of inertial to viscous forces.
REYNOLDS STRESSESAdditional terms produced by time averaging the turbulent Navier   Stokes equations; physically, the nine stress components associated with turbulent transport of momentum.
REYNOLDS TRANSPORT THEOREMReynolds transport theorem describes the relationship between Eulerian and Lagrangian frames of reference.
REYNOLDS-AVERAGINGEnsemble averaging to remove small scale unsteadiness to enable the simulation of turbulent flows.
RICHARDSON EXTRAPOLATIONA method of approximating a variable value using several different grids and an error estimate.
RICHARDSON NUMBERA stability criterion for stratified flows.
RICHARDSON'S METHODAn extrapolation method for improving approximate finite-difference results without the explicit use of a difference correction.
RICHTMYER ALGORITHMA two-step scheme that avoids estimation of a Jacobian matrix.
RIEMANN PROBLEMA system containing discontinuous neighbouring states e.g. a shock problem.
RIGID BODY DEFORMATIONSA non-zero displacement pattern that has zero strain energy associate with it.
RIGID BODY DISPLACEMENTA non-zero displacement pattern that has zero strain energy associate with it.
RIGID BODY MODESIf a displaced shape does not give rise to any strain energy in the structure then this a rigid body mode. A general three dimensional unsupported structure has 6 rigid body modes, 3 translation and 3 rotation.
RIGID LINKS RIGID OFFSETSThis is a connection between two non-coincident nodes assuming that the connection is infinitely stiff. This allows the degrees of freedom at one of the nodes (the slave node) to be deleted from the system. It is a form of multi-point constraint.
RNG K- TURBULENCE MODELA variant of the standard k-  turbulence  model where the model constants are derived from Renormalisation Group theory and are based on statistical techniques as opposed to empirical techniques as used in the standard k- model.
ROBIN BOUNDARY CONDITIONA linear combination of a variable’s specified value and its normal derivative.
ROE LINEARISATIONA method of extending the linear wave decomposition to non-linear equations.
ROE'S APPROXIMATE RIEMANN SOLVERRoe linearisation of the conserved flux Jacobians applied to hyperbolic equations.
ROTATING FRAME OF REFERENCEA physical coordinate system that rotates at constant angular velocity in order to observe fluid motion relative to an object rotating at (usually) the same angular velocity. Flow equations and calculations employ extra terms to accommodate the effects of rotation. Usually applied to turbomachinery flows.
ROTHALPYRotational stagnation enthalpy. The total energy content in a steadily rotating frame of reference.
ROUNDOFF ERRORComputers have a fixed wordlength and hence only hold numbers to a certain number of significant figures. If two close numbers are subtracted one from another then the result loses the first set of significant figures and hence loses accuracy. This is round off error.
ROUND-OFF ERRORAn error caused by the storage of a real number using a restricted number of digits, rounding off to the nearest value.
ROW VECTOR ROW MATRIXA 1xn matrix written as a horizontal string of numbers. It is the transpose of a column vector.
RSM (REYNOLDS STRESS MODEL)A closure turbulence model with six equations for Reynolds stress transport and the scalar dissipation rate.
RUNGE-KUTTA SCHEMEAn explicit non-linear time integration technique.
RUNGE-KUTTA TIME STEPPINGA method of stabilising a higher order Runge-Kutta  scheme, also known as a multistage method.
RUPTURE TIMEThe time required for a structure to fail due to continuous creep deformation.
SAFE LIFEA design philosophy in which products are designed to survive a specific operational life with a chosen reserve.
SANDWICH STRUCTUREA composite structure composed of lightweight core material (usually honeycomb or foam) to which two relatively thin, dense, high-strength, functional or decorative skins are adhered.
SCALAR CONTROL VOLUMEThe control volume containing the scalar variables in a staggered grid arrangement.
SCALAR FLUXRate of flow of a scalar quantity per unit area.
SCALARS VECTORSQuantities that have no direction associated with them, e.g. temperatures. Scalar problems only have one degree of freedom at a node. Vector quantities have a direction associated with them, e.g. displacements. Vector problems have more than one degree of freedom at a node.
SCALE SIMILARITY MODELA sub-grid scale model for the behaviour of turbulent eddies.
SCHMIDT NUMBERA dimensionless number that is the ratio of kinematic viscosity to diffusivity and is the analogue for the diffusion of chemical species to the Prandtl  number for the diffusion of heat.
SCHWARTZ-CHRISTOFFEL TRANSFORMATIONFormula to produce conformal mapping of a closed region in the physical plane to the upper half of the transform plane.
SECANT STIFFNESSThe stiffness defined by the slope of the line from the origin to the current point of interest on a load/deflection curve.
SECOND ORDER SCHEMEA scheme which is second-order accurate in terms of a Taylor series.
SECOND PIOLA-KIRCHHOFF STRESSThe work conjugate stress measure to the Green strain.
SECONDARY COMPONENTSComponents of a structure not of direct interest but they may have some influence of the behaviour of the part of the structure that is of interest (the primary component) and have to be included in the analysis in some approximate form.
SECONDARY CREEPThat part of a creep test where the strain rate is constant.
SECONDARY FLOWSFlows in a transverse plane to the main streamwise flow.
SEEPAGE FLOWFlows in porous materials
SEGREGATED SOLVERA solver in which the governing equations are segregated from one another and solved sequentially. This approach is often used for incompressible flows. It is an alternative to a coupled solver.
SEISMIC ANALYSISThe calculation of the dynamic displacement and stress response arising from earthquake exicitations.
SELECTED REDUCED INTEGRATIONA form of Gaussian quadrature where different sets of Gauss points are used for different strain components.
SELF ADJOINT EQUATIONSA form of matrix products that preserves symmetry of equations. The product A*B*A(transpose) is self-adjoint if the matrix B is symmetric. The result of the product will be symmetric for any form of A that is of a size compatible with B. This form of equation occurs regularly within the finite element method. Typically it means that for a structural analysis the stiffness (and mass) matrices for any element or element assembly will be symmetric.
SELF EQUILIBRATING LOADSA load set is self equilibrating if all of its resultants are zero. Both translation and moment resultants are zero.
SELF PRESERVATIONA flow where functions of flow variables become invariant with distance in the flow direction.
SEMI-IMPLICITA method of solution which is a mixture of an explicit and a fully implicit method.
SEMI-LOOF ELEMENTA form of thick shell element.
SHAKEDOWNOccurs in cyclic loading where the plastic strain in each cycle stabilises so that the total strain within a cycle is less than twice the yield strain (the strain when the stress reaches the yield stress).
SHALLOW WATER EQUATIONSEquations that describe the time-dependent and spatial distribution of the height of a free surface in a stream with velocity.
SHAPE FUNCTIONSEquations which are used to define the variation of the geometry and main degrees of freedom (typically displacement) within an element: the equations vary over different element types.
SHAPE PARAMETERSWays of defining an element’s shape, with particular reference to how the shape differs from the theoretically perfect shape for that element type. Parameters include aspect ratio (q.v.), taper, skew, curvature, warpage and variation of the Jacobian (q.v.).
SHEAR LOCKINGThe phenomena which occurs when thick elements give overstiff results when modelling thin beams/plates/shells, due to an excess of shear energy being present. It can also affect 2D and 3D continuum elements.
SIMPLE ALGORITHM(Semi-Implicit Method for Pressure-Linked Equations) an algorithm which is used to compensate for the lack of an explicit pressure equation in the Navier  Stokes equations using an iterative procedure consisting of a predictor  and  a  corrector step.
SIMPLEC ALGORITHMThe basic SIMPLE algorithm can suffer from slow convergence properties and can, in certain circumstances, also suffer from a lack of robustness and over-sensitivity to under-relaxation  parameters. To overcome these limitation several variants of the SIMPLE algorithm have been derived. SIMPLEC is one of these variants.
SIMPLER ALGORITHM(SIMPLE Revised) the basic SIMPLE algorithm can suffer from slow convergence properties and can, in certain circumstances, also suffer from a lack of robustness and over-sensitivity to under-relaxation parameters. To overcome these limitations several variants of the SIMPLE algorithm have been derived. SIMPLER is one of these variants.
SIMPSONS RULEA method for numerically integrating a function.
SIMULTANEOUS VECTOR ITERATIONA method for finding the first few eigenvalues and eigenvectors of a finite element system. This is also known as subspace vector iteration.
SINGLE DEGREE OF FREEDOMThe system is defined by a single force/displacement equation.
SINGLE ELEMENT TESTSAny test of an element’s performance using only one element (q.v. also patch test and continuum region element (CRE) method).
SINGLE POINT CONSTRAINTWhere the constraint is unique to a single node point.
SINGULAR MATRIXA square matrix that cannot be inverted.
SINGULARITY METHODA technique to solve a linear Laplace  equation. A linear superposition of known elementary flow fields, such as vortex and source singularities, is defined. The unknown coefficients of this linear superposition are obtained by stipulating that the resultant velocity field satisfies the condition of vanishing normal velocity along solid body surfaces.
SINKNegative source term.
SIP (STRONGLY IMPLICIT PROCEDURE)A technique for solving simultaneous equation sets, also known as Stone’s Method. Alternative techniques include TDMA, Gauss-Siedel, conjugate gradient.
SKEW DISTORTION (ANGULAR DISTORTION)A measure of the angular distortion arising between two vectors that are at right angles in the basis space when these are mapped to the real coordinate space. If this angle approaches zero the element becomes ill- conditioned.
SKEW UPWIND SCHEMEA higher order discretisation  scheme where the interface value of the dependent variable is established by the upstream conditions in the flow direction. Often accurate but can produce non physical under- or overshoots in the regions of steep gradients.
SKEWNESSA non-dimensional parameter which characterises the extent to which a cell is deformed from an equilateral cell of equivalent volume and the same basic shape (triangle, square, cube, etc.).
SKIN FRICTION COEFFICIENTA non-dimensional parameter that characterises the viscous friction forces of the flow over a solid surface.
SLIDINGIn contact analysis, when adjacent surfaces move tangentially to one another.
SMEARED CRACK MODELIn the non-linear analysis of concrete structures, a model which does not follow discrete cracks, but assumes damage is caused by closely spaced cracks associated with an integration point.
SNAP BACK (ALSO CALLED SNAP THROUGH)A situation that occurs when a vertical line in the load-displacement curve is encountered, and two or more equilibrium states are possible for the same applied load.
SOFTENINGIn plastic flow, this is a contraction of the yield surface that leads to localisation phenomena.
SOLID ELEMENTSThree dimensional continuum elements.
SOLUTION ACCURACYThe accuracy of the solution of the equations used in the finite element method, usually referring to the main stiffness equations. When a very large number of variables exist or the model generated is poor, accuracy can be lost due to ill-conditioning arising from the numerical processes.
SOLUTION ADAPTIVE MESHA CFD grid that automatically adjusts to the emerging CFD solution. It has the substantial advantage that steep gradients of dependent variables can be resolved with a locally refined grid, which does not have to be fixed in advance of starting the simulation. Solution adaptive grids are often used in the capture of sharp flow features such as shocks or moving deflagration fronts.
SOLUTION DIAGNOSTICSMessages that are generated as the finite element solution progresses. These should always be checked for relevance but the are often only provided for information purposes
SOLUTION EFFICIENCYAn indication of the efficiency of the solution of the equations used in the finite element method, usually referring to the main stiffness equations. Minimising the number of such equations without compromising solution accuracy is a common challenge.
SOLUTION OF ALGEBRAIC EQUATIONSCFD simulations are based on the solution of some form of the governing Navier- Stokes  equations. These are highly non-linear partial differential equations that cannot, except in a few trivial cases, be solved analytically. Typically, the partial differential equations are discretised and rearranged to form a set of algebraic equations, essentially consisting of a large set of simultaneous equations. Solution of the algebraic equations provides an approximate discrete solution of the governing flow equations.
SOLUTIONS VECTORA vector of dependent flow variables. Usually an expression confined to the external aerodynamics CFD community. As an example, the solution vector for the Euler equations consists of the three components of velocity, pressure and internal energy.
SOURCE TERMSTerms which appear in the general conservation equation of a variable and which cannot be accommodated in the unsteady, convective or diffusive terms. They are meant primarily for internal generation processes such as heat generation in a fluid, production of a chemical species in a reaction, and the generation of turbulent kinetic  energy. However when the corresponding physical quantity is destroyed rather than produced, the source term becomes negative and may be known as a sink term.
SPACE MARCHINGEarly CFD methods were often limited, by restricted RAM, to the solution of a parabolic form of the governing equations using a space-marching method. In this technique the solution is marched downstream, with no upstream influence of downstream conditions allowed. This permits one to effectively solve 3-D problems by storing only 2-D arrays, and 2-D problems by storing only 1-D arrays. Attached boundary layer and supersonic flows are typical candidates for space- marching approaches.
SPARSE MATRIX METHODSSolution methods that exploit the sparse nature of finite element equations. Such methods include the frontal solution and Cholesky (skyline) factorisation for direct solutions, conjugate gradient methods for iterative solutions and the Lanczos method and subspace iteration (simultaneous vector iteration) for eigenvalue solutions.
SPECTRAL DENSITYThe Fourier transform of the correlation function. In random vibrations it gives a measure of the significant frequency content in a system. White noise has a constant spectral density for all frequencies.
SPECTRAL METHODA method that uses the Fast Fourier Transform or similar polynomial method to solve Navier-Stokes  equations or other partial differential equations. Commonly used for Direct Numerical Simulations. Spectral methods are higher order methods, of the N-th order if there are N grid points.
SPLINE CURVESA curve fitting technique that preserves zero, first and second derivative continuity across segment boundaries.
SPLINE METHODSAn implicit finite difference relationship for the first and second derivative derived from the Taylor series expansions of the transport equations. Spline methods have been used extensively in Finite  Element  codes, but have not been found to be advantageous for finite volume / finite difference codes.
SPLITTING METHODSSee approximate factorisation technique
SPURIOUS CRACKSCracks that appear in a mesh when the elements are not correctly connected together. This is usually an error in the mesh generation process.
SPURIOUS OSCILLATIONSUnphysical oscillations of a solution generated by the discretisation scheme.
STABILITYThe property of a numerical method that progresses towards a solution without wild oscillations or divergence.
STABILITY ANALYSISA mathematical procedure examining the behaviour of a discretisation scheme and providing criteria for its stability.
STABILITY CRITERIONFor a given discretisation scheme, a stability criterion provides the conditions, for instance, on time-step or space discretisation, to achieve convergence.
STAGGERED GRIDIn a staggered grid, the velocity components are calculated at the points that lie on the faces of the control volumes while all the other variables are calculated at the centre of the control volumes. A grid is forward or backward staggered depending on whether the staggered grid is offset forwards or backwards.
STATICALLY DETERMINATE STRUCTUREA structure where all of the unknowns can be found from equilibrium considerations alone.
STATICALLY EQUIVALENT LOADSEquivalent nodal loads that have the same equilibrium resultants as the applied loads but do not necessarily do the same work as the applied loads.
STATICALLY INDETERMINATE STRUCTURE REDUNDANTA structure where all of the unknowns can not be found from equilibrium considerations alone. The compatibility equations must also be used. In this case the structure is said to be redundant.
STATIONARY RANDOM EXCITATIONA force or response that is random but its statistical characteristics do not vary with time.
STEADY STATE CREEP LAWA creep model in which there are no hardening or softening effects.
STEADY STATE FLOWA flow field that is independent of time.
STEADY STATE RESPONSEThe response of the system to a periodic forcing function when all of the transient components of the response have become insignificant.
STEEPEST DESCENT METHODSA method for finding the minimum value of a function.
STEGER-WARMING FLUX SPLITTINGAn upwind discretisation scheme that splits the fluxes according to the signs of the eigenvalues. This scheme aims to capture discontinuities.
STENCILA representation of a difference formula, based upon the values at neighbouring points.
STEPAn international standard for the exchange of CAD / CAM data (ISO 10303, The STandard for the Exchange of Product model information).
STEP SIZESpatially, the step size is the cell size. Temporally, the step size is the size of the time increments.
STEP-BY-STEP INTEGRATIONMethods of numerically integrating time varying equations of motion. These methods can be either explicit or implicit.
STIFF PROBLEMSStiff problems occur when there are two or more very different scales of the independent variables on which the dependent variables are changing.
STIFFNESS MATRIXThe parameter(s) that relate the displacement(s) to the force(s). For a discrete parameter multi degree of freedom model this is usually given as a stiffness matrix.
STOKES EQUATIONSFluid flow equations where convection terms are neglected with respect to viscous terms.
STOKES’ HYPOTHESISAn approximation that can be applied when the Reynolds  number is small compared to one, i.e. for strongly viscous-dominated flows. It neglects convection terms with respect to viscous terms and results in the Stokes equations.
STRAIN ENERGYThe energy stored in the system by the stiffness when it is displaced from its equilibrium position.
STRAIN ENERGY RELEASE RATEFor a hypothetically small increase in crack length or area, this is the amount of strain energy released divided by that length or area. It equals the negative of the potential energy release rate (q.v.) when elastic conditions predominate.
STRAIN HARDENING LAWUsed in analysing creep behaviour under variable load where the creep strain rate is assumed to depend on the current stress and accumulated creep strain, or in plasticity where the current yield stress is a function of the plastic strain.
STRAIN-LIFE APPROACHIn fatigue, a method whereby the predicted life of a product is based on calculated strain values, typically used in low cycle fatigue.
STREAM FUNCTIONThe mathematical description of two-dimensional flows that allows the velocity field to be represented in terms of a single function  such that v = -grad .
STREAMLINEAn imaginary instantaneous line, which characterises a flow such that, at every point along the line, the velocity vector is tangent to the line. For steady  flow, streamlines and path lines are identical.
STREAMLINE CO-ORDINATESA co-ordinate system fitted to the flow such that a co-ordinate direction is aligned with the flow streamlines.
STREAMLINE UPWIND SCHEMEA scheme used to stabilise the higher order symmetric operators from the Galerkin method by adding numerical  diffusion in the streamwise direction only, thus preserving the accuracy of the Galerkin method in the cross-stream direction. This scheme is often used in Finite  Element Analysis and for convection dominated flows.
STRESS AVERAGING STRESS SMOOTHINGThe process of filtering the raw finite element stress results to obtain the most realistic estimates of the true state of stress.
STRESS CONCENTRATIONA local area of the structure where the stresses are significantly higher than the general stress level. A fine mesh of elements is required in such regions if accurate estimates of the stress concentration values are required.
STRESS DISCONTINUITIES STRESS ERROR ESTIMATESLines along which the stresses are discontinous. If the geometry or loading changes abruptly along a line then the true stress can be discontinous. In a finite element solution the element assumptions means that the stresses will generally be discontinuous across element boundaries. The degree of discontinuity can then be used to form an estimate of the error in the stress within the finite element calculation.
STRESS EXTRAPOLATIONThe process of taking the stress results at the optimum sampling points for an element and extrapolating these to the element node points.
STRESS INTENSITY FACTORA fracture parameter at a crack tip when under conditions of LEFM. It is a function of applied load and crack length, suitably dimensioned to have a finite value at the tip even though the stresses are singular there, and may be used to characterise the state of fracture there.
STRESS RELAXATIONOccurs in creep problems when the structure is loaded up to a certain stress level and then held at constant strain.
STRESS SUBSTITUTION METHODA method of calculating the stress intensity factor at a given crack tip using the local stresses from FE analysis and known crack tip equations.
STRESS VECTOR STRESS TENSOR STRAIN VECTOR STRAIN TENSORThe stress (strain) vector is the components os stress (strain) written as a colunn vector. For a general three dimensional body this is a (6x1) matrix. The components of stress (strain) written in tensor form. For a general three dimensional body this forms a (3x3) matrix with the direct terms down the diagonal and the shear terms as the off-diagonals.
STRESS WAVESElastic stresses that propagate through materials at high speeds due to impact loads.
STRESS-LIFE APPROACHIn fatigue, a method whereby the predicted life of a product is based on calculated stress values, typically at stress concentrations and for high cycle fatigue.
STRESS-STRAIN LAWThe material property behaviour relating stress to strain. For a linear behaviour this is Hookes law (linear elasticity). For elastic plastic behaviour it is a combination of Hookes law and the Prandtl-Reuss equations.
STRETCHING FUNCTIONA stretching function is used to define how the separation of grid lines varies. If there is no stretching, the grid is uniform. However, a non-uniform grid is often needed to optimise the number of cells used. In this case, a stretching function may be used to fix the disposition of the grid lines.
STROUHAL NUMBERA dimensionless number used to characterise the periodicity of unsteadiness occurring in flows exhibiting a dominant frequency of unsteady behaviour. The number is calculated from the product of frequency and representative dimension of an object immersed in the flow, divided by the free stream velocity. In physical terms, it represents the ratio of the time of transit of the free stream fluid past the object, to the period of the dominant unsteadiness.
STRUCTURED GRIDA grid in which the cells (hexahedra in three dimensions or quadrilaterals in two dimensions) form a regular pattern. The grid lines are continuous across the domain and are usually aligned with the co-ordinate directions or mirror the boundary topography. Each grid cell in a structured grid can thus be defined by a matrix of two or three numbers representing positions along a grid line in each co- ordinate direction.
STRUCTURED GRID (OR MESH)A grid (in CFD) or mesh where the elements form a regular pattern.
SUBGRID SCALEAn effect or geometrical entity that is smaller than the size of a single grid cell.
SUBSONIC FLOWFlow that is slower than the speed of sound, i.e. the Mach number is less than unity.
SUBSPACE VECTOR ITERATIONA method for finding the first few eigenvalues and eigenvectors of a finite element system. This is also known as simultaneous vector iteration.
SUBSTANTIAL DERIVATIVEPhysically the average time rate of the change of a variable.
SUBSTRUCTURE (ALSO CALLED SUPERELEMENT)A mesh modelling technique whereby a part of a structure, containing a number of elements, can be stored by the software as a single element. It can then be used for a variety of different purposes, just as if it were a new element type with its own stiffness matrix.
SUBSTRUCTURING SUPER ELEMENT METHODSubstructuring is a form of equation solution method where the structure is split into a series of smaller structures - the substructures. These are solved to eliminate the internal freedoms and the complete problem solved by only assembling the freedoms on the common boundaries between the substructures. The intermediate solution where the internal freedoms of a substructure have been eliminated gives the super element matrix for the substructure.
SUCCESSIVE OVER-RELAXATION (SOR)A method of solving matrices.
SUPERFICIAL VELOCITYA velocity in a porous medium where the fluid volume is not reduced to take account of the degree of blockage; i.e. the velocity that would occur if the mass flux of fluid was distributed over the entire area occupied by fluid and solid.
SUPERPOSITIONFor a linear system the response is the same if it is found by adding together two or more separate forcing functions and then solving the equations or by solving for the separate forcing functions and then adding the responses together. The second method of solving for each forcing function and adding the response is superposition. A modal solution and a Fourier series solution both imply superposition.
SUPERSONIC FLOWFlow that is faster than the speed of sound, i.e. the Mach number is greater than unity.
SUPPORTSDegrees of freedom where the variable is known before the solution is found. Typically the zero displacements at fixed points in a structural analysis or the points of known temperature in a heat conduction analysis. Generally there must be some points of known value (i.e the structure must be supported) before the equations can be solved.
SURFACE ELEMENTSpecial elements that are used to model surface boundary conditions. Typically surface heat transfer elements used to model surface heat transfer coefficients in heat conduction problems.
SUTHERLAND’S FORMULAA formula for the dynamic viscosity as a function of temperature using a constant known as the Sutherland constant.
SWEEP DIRECTIONThe direction in which the matrix is solved.
SYMMETRICAL MATRIX SKEW SYMMETRIC MATRIX HERMITIAN A matrix is symmetric if it is square and if the ij term is equal to the ji term. A matrix is SKEW symmetric if it is square and if the ij term is equal to minus the ji term. All of the diagonal terms are zero. A matrix is Hermitian if it is square, the real part is symmetric and the imaginary part is skew symmetric.
SYMMETRYA structural problem is symmetric if one half of the structure and the loading is the mirror image of the other half. Symmetry can be used to half the problem size.
SYMMETRY (OF MODEL)In constructing a finite element model, the meshing of similar shapes with similar loading within the model can be avoided by using the principles of symmetry, and by using suitable boundary conditions. The different types of symmetry include: repetitive, mirror, axial and cyclic. Asymmetric loading can also be modelled with suitable boundary conditions.
SYMMETRY BOUNDARY CONDITIONBoundary condition where the normal velocity is zero and the normal gradients of all other variables are also zero.
TANGENT STIFFNESSFor non-linear problems this is the slope of the load/deflection curve for the current solution position.
TANGENT STIFFNESS MATRIXThe matrix of coefficients corresponding to the derivatives of the residual forces with respect to the displacement degrees of freedom: this matrix is evaluated and factorised during the incremental-iterative solution procedure.
TDMASee tri-diagonal matrix algorithm.
TERTIARY CREEPThat part of a creep test where the strain rate is increasing.
TETRAHEDRAL ELEMENTS3D computational cells that are tetrahedral in shape - i.e. have four sides.
TETRAHEDRON TETRAHEDRAL ELEMENTA three dimensional four sided solid element.
THERMAL CAPACITYThe material property defining the thermal inertia of a material. It relates the rate of change of temperature with time to heat flux.
THERMAL CONDUCTIVITYThe material property relating temperature gradient to heat flux.
THERMAL CONTACTThe analysis of contacting surfaces when thermal effects are significant.
THERMAL LOADSThe equivalent loads on a structure arising from thermal strains. These in turn arise from a temperature change.
THERMAL STRAINSThe components of strain arising from a change in temperature.
THERMALLY PERFECT GASA gas for which (pV)/(mT) is constant.
THIN SHELL ELEMENT THICK SHELL ELEMENTIn a shell element the geometry is very much thinner in one direction than the other two. It can then be assumed stresses can only vary linearly at most in the thickness direction. If the through thickness shear strains can be taken as zero then a thin shell model is formed. This uses the Kirchoff shell theory If the transverse shear strains are not ignored then a thick shell model is formed. This uses the Mindlin shell theory. For the finite element method the thick shell theory generates the most reliable form of shell elements. There are two forms of such elements, the Mindlin shell and the Semi-Loof shell.
THOMAS ALGORITHMSee tri-diagonal matrix algorithm.
TIME DOMAINThe structures forcing function and the consequent response is defined in terms of time histories. The Fourier transform of the time domain gives the corresponding quantity in the frequency domain.
TIME HARDENING LAWUsed in analysing creep behaviour under variable load where the creep strain rate is assumed to depend on the current stress and time from the start of the test.
TIME MARCHINGA solution technique to obtain a steady state solution by solving transiently until the rate of change from one time step to the next is negligible.
TIME STEPThe incremental change in time for which a flow is being solved.
TIME STEPPING SCHEMESMethods for integrating the governing equations of time dependent non-linear problems. Examples include Newmark’s family of methods for solving the transient dynamic equilibrium equations and time marching procedures for creep analysis.
TOTAL LAGRANGIAN FORMULATIONIn geometrically non-linear analysis, a formulation in which all static and kinematic variables are referred to the initial undeformed configuration (see also updated Lagrangian formulation).
TOTAL PRESSUREThe static pressure plus the dynamic pressure or the pressure obtained by bringing a fluid to rest isentropically.
TOTAL VARIATION DIMINISHING (TVD) SCHEMESA higher order differencing scheme.
TRACE OF THE MATRIXThe sum of the leading diagonal terms of the matrix.
TRANSFINITE INTERPOLATIONAn algebraic method of interpolating a mesh.
TRANSFINITE MAPPINGA systematic method for generating element shape functions for irregular node distributions on an element.
TRANSFORMATION METHODSolution techniques that transform coordinate and force systems to generate a simpler form of solution. The eigenvectors can be used to transform coupled dynamic equations to a series of single degree of freedom equations.
TRANSIENTSee unsteady.
TRANSIENT ANALYSISAn analysis is transient when at least one of the parameters involved in the boundary conditions, material properties or loading conditions is time dependent.
TRANSIENT FORCEA forcing function that varies for a short period of time and then settles to a constant value.
TRANSIENT RESPONSEThe response of a system to applied forces that are of short duration compared to the periods of the resonant frequencies of the system.
TRANSITION ELEMENTSpecial elements that have sides with different numbers of nodes. They are used to couple elements with different orders of interpolation, typically a transition element with two nodes on one edge and three on another is used to couple a 4-node quad to an 8-node quad.
TRANSITIONAL FLOWFlow which changes from exhibiting laminar behaviour to turbulent behaviour
TRANSONIC FLOWFlow that changes from subsonic to supersonic or vice versa.
TRANSPORT EQUATIONA differential equation describing the redistribution of a property or quantity through a medium or through space.
TRANSPORTIVENESSA property of the numerical scheme that accounts for the direction in which the relative strengths of convection and diffusion influence the flow.
TRESCA YIELD CRITERIONIs used for metals and assumes that yielding starts when the maximum value of the shear stress reaches a given value. It gives a hexagonal cylindrical shape in principal stress space.
TRIANGULAR ELEMENTA two dimensional computational cell that is triangular in shape.
TRIANGULAR ELEMENTSTwo dimensional or surface elements that have three edges.
TRI-DIAGONAL MATRIX ALGORITHM (TDMA)A particularly efficient method used to solve the matrix equation set Ax = b, where A is such that all non-zero coefficients align themselves along three diagonals.
TRUE STRAIN (ALSO CALLED LOGARITHMIC STRAIN OR NATURAL STRAIN)A particular strain measure used in large strain elasto-plasticity; the log of one plus the engineering strain, or the integral of the incremental change of length over the current length.
TRUE STRESS (ALSO CALLED CAUCHY STRESS)The force divided by the current (instantaneous) area.
TRUNCATION ERRORThe result of the truncation of the expansion series used in the discretisation scheme.
TURBULENCEA chaotic state of fluid motion where the velocity and pressure at a point change continuously with time.
TURBULENCE CHARACTERISTIC LENGTHA typical dimension of a turbulent eddy.
TURBULENCE MODELSSets of equations that determine the turbulent transport terms (Reynolds stresses) in the mean flow equations. They are based on hypotheses about turbulent processes and generally require significant empirical input in the form of constants or functions. These time averaged models do not simulate the details of the turbulent motion (the turbulent eddies), only the effect of turbulence on the mean flow behaviour. Thus, with a particular set of empirical constants, they are valid only for a certain flow or at most a range of flows. This is also known as a RANS approach (Reynolds Averaged Navier-Stokes).
TURBULENCE PRODUCTIONThe generation of turbulence.
TURBULENCE SPECTRUMThe distribution of eddy scales from smallest to largest which are present in a flow.
TURBULENT DISSIPATIONThe reduction in turbulent kinetic energy caused by the work done by the smallest eddies converting turbulent kinetic energy to thermal internal energy.
TURBULENT ENERGYSee turbulent kinetic energy.
TURBULENT FLUXTransport of a quantity associated with turbulent motion.
TURBULENT KINETIC ENERGYThe kinetic energy associated with the turbulent fluctuations in velocity.
TURBULENT LENGTH SCALEThe length scale characteristic of the largest eddies which contain most of the turbulent kinetic energy.
TURBULENT PRANDTL NUMBERIn the "eddy  viscosity  model" of turbulence, transport of momentum due to turbulence is modelled by adding an effective viscosity representative of local turbulence conditions (the eddy viscosity) to the true fluid viscosity in the diffusion terms of the momentum equations. By analogy, transport of heat due to turbulence is modelled by adding an effective thermal diffusivity to the true fluid thermal diffusivity in the diffusion terms of the energy equation. The turbulent Prandtl number is the ratio of the eddy viscosity to this effective thermal diffusivity.
TVD SCHEMESSee Total Variation Diminishing schemes
TWO-EQUATION MODELA turbulence  model that uses two transport  equations to model the effects of turbulence in the RANS equations.
TWO-LEVEL SCHEMEA temporal scheme that stores variables at two time levels.
ULTIMATE STRESSThe failure stress (or equivalent stress) for the material.
UNCERTAINTY QUANTIFICATIONFormulation of a statistical model to characterise imperfect and/or unknown information in engineering simulation and physical testing for predictions and decision making.
UNCONDITIONAL INSTABILITYThe property of a scheme which is always unstable, regardless of values of parameters such as cell size or size of time steps.
UNCONDITIONAL STABILITYThe property of a scheme that is always stable, i.e. no constraints exist on parameters such as cell size or time step size.
UNCOUPLED PARTICLE FLOWFlow of discrete particles (bubbles or drops) in a continuum in which the movement of the particles does not influence the flow of the continuum.
UNDAMPED NATURAL FREQUENCYThe square root of the ratio of the stiffness to the mass (the square root of the eigenvalue). It is the frequency at which an undamped system vibrates naturally. A system with n degrees of freedom has n natural frequencies.
UNDER DAMPED SYSTEMA system that has an equation of motion where the damping is less than critical. It has an oscillatory impulse response.
UNDERRELAXATIONAn algorithm restraining the amount by which a variable may change from one iteration to the next.
UNIFORM GRIDA computational grid in which each cell is the same size and shape.
UNIT MATRIXA diagonal matrix with unit values down the diagonal.
UNSTABLE SCHEMEA scheme which does not exhibit stability, i.e. it does not converge.
UNSTEADY FLOWFlow which changes with time.
UNSTRUCTURED GRIDA grid in which the cells form no regular pattern. Unstructured grids allow highly complex geometries to be modelled with relative ease compared to structured grids and allow for greater cell concentrations in regions of flow complexity.
UNSTRUCTURED GRID (OR MESH)A grid (in CFD) or mesh where the elements form no regular pattern.
UPDATED LAGRANGIAN FORMULATIONIn geometrically non-linear analysis, a formulation in which all static and kinematic variables are referred to the last calculated configuration (see also total Lagrangian formulation).
UPWIND DIFFERENCING SCHEMEA discretisation  scheme that uses the upstream variable values. Also known as donor cell differencing.
UPWIND FORMULATIONSee upwind differencing scheme.
UPWINDING IN FLUIDSA special form of weighting function used in viscous flow problems (solution to the Navier-Stokes equations) used in the weighted residual method to bias the results in the direction of the flow.
URANSUnsteady Reynolds Averaged Navier Stokes. See RANS.
VALIDATIONThe process of determining how accurately a simulation represents the real world. cf Verification.
VAN LEER’S FLUX SPLITTINGBasically a technique for discretising convective terms, sometimes called the MUSCL  scheme. Alternative approaches include QUICK (a form of upwinding) and central differences. MUSCL includes a free parameter and for certain values, MUSCL reduces to QUICK or second order central differencing.
VARIABLE BANDWIDTH (SKYLINE)A sparse matrix where the bandwidth is not constant. Some times called a skyline matrix.
VARIATIONAL FORMULATIONA minimalisation formulation used in the finite  element  method, especially for structural analysis.
VECTOR PLOTSA method of displaying a vector quantity at discrete grid locations, using arrows to illustrate both magnitude and direction.
VELOCITYThe first time derivative of the displacement.
VELOCITY CORRECTIONUsed in pressure  correction  methods, such as SIMPLE, to correct for guessed velocity values.
VELOCITY DEFECT LAWA law that treats the wall shear stress as the cause of a defect, which decreases with distance from the wall.
VELOCITY PROFILESSectional variation in velocity, e.g. parabolic variation in fully developed laminar pipe flow.
VERIFICATIONThe process of determining if a simulation accurately represents the conceptual model. A verified simulation does not make any claim relating to the representation of the real world by the simulation. cf. Validation
VERTEX CENTREDA formulation in which cell vertices are located mid-way between cell centres.
VERTEX-BASED FORMULATIONA formulation in which the variable values are stored at the cell vertices.
VIRTUAL CRACK EXTENSION METHODA method for calculating fracture criteria at a crack tip using the potential energy change with crack growth and utilising in an efficient manner certain characteristics of the stiffness equations.
VIRTUAL DISPLACEMENTSAn arbitrary imaginary change of the system configuration consistent with its constraints.
VIRTUAL WORK VIRTUAL DISPLACEMENTS VIRTUAL FORCESTechniques for using work arguments to establish equilibrium equations from compatibility equations (virtual displacements) and to establish compatibility equations from equilibrium (virtual forces).
VISCO-ELASTICITYA non-linear material behaviour in which both the effects of elasticity and creep are exhibited, so that the stress is dependent on the strain rate.
VISCO-PLASTICITYA non-linear material behaviour theory in which time rate effects are included in the plastic deformation process; thus, stresses and strains describing the plastic state are also time dependent.
VISCOSITYThe resistance of a fluid to shear; relating shear stress to the rate of angular deformation of fluid elements.
VISCOUS DAMPINGThe damping is viscous when the damping force is proportional to the velocity.
VISCOUS DAMPING MATRIXThe matrix relating a set of velocities to their corresponding velocities
VISCOUS DISSIPATIONThe dissipation of turbulent kinetic energy caused by work done by the smallest eddies against viscous stresses.
VISCOUS INTERACTIONSee viscous-inviscid interaction.
VISCOUS STRESSESStresses due to the resistance of relative movement of fluid layers either past one another or other fluids or solids. They are generally the dominant forces in near wall regions.
VISCOUS SUB-LAYERThe region close to a wall in which the viscous forces dominate the flow.
VISCOUS-INVISCID INTERACTIONA flow field in which significant interaction takes place between a growing boundary layer and the adjacent inviscid flow.
VOLUME DISTORTION VOLUMETRIC DISTORTIONThe distortion measured by the determinant of the Jacobian matrix, det j.
VOLUME-OF-FLUID METHODA multiphase (multi-fluid) technique in which a single set of momentum equations is shared by the fluids and the volume fraction in each cell is tracked through the domain. This method is generally used where the interface between the fluids is of interest.
VOLUMETRIC STRAINSee hydrostatic strain.
VOLUMETRIC STRESSSee hydrostatic stress.
VON KARMAN CONSTANTThe constant used in a semi-empirical relationship developed by Theodore von Karman to relate turbulent mixing length to velocity gradient. Most commonly encountered in CFD in the formulation of wall  functions for turbulent boundary layers.
VON MISES EQUIVALENT STRESS TRESCA EQUIVALENT STRESSEquivalent stress measures to represent the maximum shear stress in a material. These are used to characterise flow failures (e.g. plasticity and creep). From test results the Von-Mises form seems more accurate but the Tresca form is easier to handle.
VON MISES STRESSThe second invariant of the deviatoric stress tensor. This is a scalar value and is used to relate a 3D stress field to a 1D tensile test. Thus, it is often called an effective or equivalent stress (q.v.).
VON MISES YIELD CRITERIONIs used to describe the yield of metals and assumes that yielding commences when the von Mises stress (q.v.) reaches a critical value.
VON NEUMANN STABILITY METHODA method of assessing the stability of a numerical scheme.
VORTEX METHODSMethods that simulate incompressible viscous flows using point vortices that satisfy Laplace’s equation.
VORTICITYA vector quantity that characterises the strength of rotation in a flow. The curl of velocity.