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NAFEMS Glossary of Terms D-I 

DAMAGE TOLERANCEA design and operational philosophy in which products are regularly inspected for damage, crack growth, etc., so that continued operation with the damage will not produce an imminent failure.
DAMKOHLER NUMBERIn combustion, the ratio of reagent diffusion to characteristic chemical reaction time across the flame.
DAMPED EIGENVALUESSame as complex eigenvalues.
DAMPED EIGENVECTORSSame as complex eigenvectors.
DAMPED NATURAL FREQUENCYThe frequency at which the damped system vibrates naturally when only an initial disturbance is applied.
DAMPINGAny mechanism that dissipates energy; important in dynamics analysis.
DAMPING FACTOR (DECAY FACTOR)The damping factor is the ratio of the actual damping to the critical damping. It is often specified as a percentage. If the damping factor is less than one then the system can undergo free vibrations. The free vibrations will decay to zero with time. If the damping factor is greater than one then the decay is exponential and no vibrations occur. For most structures the damping factor is very small.
DAMPING FACTOR/RATIOThe ratio of the viscous damping coefficient to the critical damping value.
DEFERRED CORRECTIONIn convective schemes, the use of higher order discretisation schemes such as QUICK may give rise to instability and unbounded solutions under some flow conditions. This is due to the appearance of negative main coefficients. To alleviate the stability problem, these schemes are formulated in a different way such that the troublesome negative coefficients are placed in the source term so as to retain positive main coefficients for the terms treated implicitly. This is known as deferred correction as the coefficients placed in the source term are treated explicitly.
DEGENERATE ELEMENTSee collapsed element.
DEGENERATE ELEMENTSElements that are defined as one shape in the basis space but they are a simpler shape in the real space. A quadrilateral can degenerate into a triangle. A brick element can degenerate into a wedge, a pyramid or a tetrahedron. Degenerate elements should be avoided in practice.
DEGREES OF FREEDOMThe number of equations of equilibrium for the system. In dynamics, the number of displacement quantities which must be considered in order to represent the effects of all of the significant inertia forces.
DELAMINATIONThe separation of layers of composites under stress.
DELTA FORMA form of writing discrete or differential equations that enables a temporal linearisation. Effectively, an efficient time integration approach.
DERIVATIVESVariables differentiated with respect to either time or space.
DET(J) DET JThe Jacobian matrix is used to relate derivatives in the basis space to the real space. The determinant of the Jacobian - det(j) - is a measure of the distortion of the element when mapping from the basis to the real space.
DETERMINISTIC ANALYSISThe applied loading is a known function of time.
DEVIATORIC STRESSA measure of stress where the hydrostatic stress has been subtracted from the actual stress. Material failures that are flow failures (plasticity and creep) fail independently of the hydrostatic stress. The failure is a function of the deviatoric stress.
DEVIATORIC STRESS AND STRAINRepresents the shear component of stress and strain, i.e. the remainder after deducting the hydrostatic component (q.v.). The deviatoric components govern plastic and creep flows, where there is change in shape but not of volume.
DIAGONAL DECAYWhen a matrix is factorised into a triangular form the ratio of a diagonal term in the factorised matrix to the corresponding term in the original matrix decreases in size as one moves down the diagonal. If the ratio goes to zero the matrix is singular and if it is negative the matrix is not positive definite. The diagonal decay can be used as an approximate estimate of the condition number of the matrix.
DIAGONAL DOMINANCESee diagonally dominant matrix.
DIAGONAL GENERALISED MATRIXThe eigenvectors of a system can be used to define a coordinate transformation such that, in these generalised coordinates the coefficient matrices (typically mass and stiffness) are diagonal
DIAGONALISATIONManipulation of a matrix to produce a diagonal matrix (a matrix in which all values are zero except those on the leading diagonal).
DIAGONALLY DOMINANT MATRIXA matrix with diagonal dominance has values on the leading diagonal that are significantly larger than those elsewhere.
DIE-AWAY LENGTHIf there is a stress concentration in a structure the high stress will reduce rapidly with distance from the peak value. The distance over which it drops to some small value is called the die-away length. A fine mesh is required over this die-away length for accurate stress results.
DIFFERENCING SCHEMEA discretisation scheme that uses the difference between known variable values to predict additional values. The higher the order of a scheme, the more accurate it is generally considered. See discretisation scheme.
DIFFERENTIAL GRID GENERATIONA grid generation method in which the mesh is generated by iteratively solving an equation set, typically the Laplace equation, which links the computational grid to the physical grid. See also algebraic grid generation.
DIFFUSIONThe natural movement of species or properties from regions of high concentration to those of lower concentration. Diffusion is modelled mathematically using Fick’s Law.
DIFFUSION COEFFICIENTA coefficient relating the rate of transport of a species or property to its concentration gradient in the carrier fluid. The value of the coefficient will depend on both the fluid and the diffusing species.
DIFFUSIVE CONDUCTANCERatio of diffusion coefficient to cell size, sometimes used in definition of cell Peclet number.
DIRECT INTEGRATIONThe name for various techniques for numerically integrating equations of motion. These are either implicit or explicit methods and include central difference, Crank-Nicholson, Runge-Kutta, Newmark beta and Wilson theta.
DIRECT METHODSSolution methods that solve a set of equations directly without the need for an iterative scheme. Also known as direct solution methods. See iterative method.
DIRECT NUMERICAL SIMULATION (DNS)A method in which the turbulent flow is directly numerically simulated without any form of time or length averaging, i.e. both the mean flow and all turbulent fluctuations (eddies) are simulated. Since turbulent eddies are both three- dimensional and unsteady (time-variant), simulations using this method must also be both three-dimensional and unsteady and, since the length and time scales of turbulent eddies cover a large range, both the grid size and the time-step size must be very small to account for the smallest fluctuations. This makes this method very computationally expensive and even with current state-of-the-art computer hardware, only practical for simple flows at low Reynolds numbers.
DIRECTION COSINESThe cosines of the angles a vector makes with the global x,y,z axes.
DIRICHLET BOUNDARY CONDITIONSA type of boundary condition where values of the flow variables are imposed on the boundaries of the flow domain.
DISCONTINUITIESSudden changes in the value of a variable. For example, shock waves.
DISCRETE CRACK MODELIn non-linear concrete analysis, a model that attempts to follow individual cracks.
DISCRETE PARAMETER MODELS (DISCRETISED APPROACH)The model is defined in terms of an ordinary differential equation and the system has a finite number of degrees of freedom.
DISCRETE PHASESecond phase in a multiphase  flow dispersed in a continuum and consisting of discrete entities such as particles, drops or bubbles.
DISCRETISATIONProcess by which the governing partial differential equations are converted into algebraic equations associated with discrete elements.
DISCRETISATION ERRORDifference between the exact solution of the governing  equations and the exact solution of the algebraic equations obtained by discretising them.
DISCRETISATION SCHEMESThe method by which the continuous variables and equations are turned into discrete variables and discrete equations.
DISPERSIVE ERRORAn error resulting from numerical dispersion.
DISPLACEMENT CONTROLWhen displacements are selected as the controlling parameter in a non-linear solution (as opposed to load or time control).
DISPLACEMENT METHOD (DISPLACEMENT SOLUTION)A form of discrete parameter model where the displacements of the system are the basic unknowns.
DISPLACEMENT PLOTSPlots showing the deformed shape of the structure. For linear small deflection problems the displacements are usually multiplied by a magnifying factor before plotting the deformed shape.
DISPLACEMENT SUBSTITUTION METHODA method of calculating the stress intensity factor at a given crack tip using the local displacements from FE analysis and known crack tip equations.
DISPLACEMENT VECTORThe nodal displacements written as a column vector.
DISSIMILAR SHAPE FUNCTIONS INCOMPATIBLE SHAPE FUNCTIONSIf two connecting elements have different shape functions along the connection line they are said to be incompatible. This should be avoided since convergence to the correct solution cannot be guarantied.
DISSIPATION ERRORAn error resulting from numerical dissipation.
DISSIPATIVE SCHEMEA scheme that artificially adds numerical dissipation.
DISTORTION (ALSO CALLED SHAPE SENSITIVITY)An indication of how much an element’s shape differs from the theoretical shape for that element type.
DISTORTION ELEMENT DISTORTIONElements are defined as simple shapes in the basis space, quadrilaterals are square, triangles are isosoles triangles. If they are not this shape in the real space they are said to be distorted. Too much distortion can lead to errors in the solution
DISTRIBUTED RESISTANCEA method for simulating a region of porous medium by the presence of a momentum sink.
DIVERGENCEThe progression of a numerical scheme away from any single answer. The opposite of convergence.
DNSSee Direct Numerical Simulation.
DOMAINThe geometrical region over which a simulation is performed. Sometimes referred to as the analytical domain or computational domain.
DOMAIN INTEGRALSUsed in fracture mechanics to evaluate fracture parameters at a crack tip, calculated using an expression integrated over an area inside a given path surrounding the tip (also called thick contours; the area form of the J-integral (q.v.)).
DOMAIN OF DEPENDENCEThe region in the x-t plane enclosed by the two characteristics and the x-axis. See region of dependence.
DOMAIN OF INFLUENCEThe region in the x-t plane bounded by the two characteristics and occurring later than the intersection point of the characteristics. See region of influence.
DONOR CELL DIFFERENCINGSee Upwind differencing.
DONOR CELL UPWINDThe upwind variable value used in upwind differencing.
DOUGLAS AND RACHFORD METHODAn Alternative Direction Implicit (ADI) method for solving the heat conduction equation in which the first step is approximated over the entire time interval and the second step is only introduced for stability reasons. It is sometimes called “Stabilisation Correction Scheme”.
DOUGLAS GUNN SPLITTING ALGORITHMAn ADI approach that is stable in three-dimensions.
DRUCKER-PRAGER EQUIVALENT STRESSESAn equivalent stress measure for friction materials (typically sand). The effect of hydrostatic stress is included in the equivalent stress.
DRUCKER-PRAGER YIELD CRITERIONAn elasto-plastic material model using both hydrostatic and deviatoric stresses, that is an inverted cone in principal stress space. Used in soil mechanics.
DUCTILE FRACTUREThis is the type of fracture occurring for a crack in a material whose behaviour is ductile, i.e. when plastic deformation is considerable. Such fracturing occurs, after some general plastic deformation as the load builds up, in metals at temperatures above the range of the brittle-ductile transition temperature.
DUPLICATE NODESMore than one node which occurs at a single geometrical location.
DYNAMIC ANALYSISAn analysis that includes the effect of the variables changing with time as well as space.
DYNAMIC BOUNDARY CONDITIONA boundary condition which changes with time.
DYNAMIC CONTACTThe analysis of contacting surfaces when inertia effects cannot be ignored.
DYNAMIC FLEXIBILITY (ALSO CALLED RECEPTANCE)The factor relating the steady state displacement response of a system to a sinusoidal force input.
DYNAMIC FLEXIBILITY MATRIXThe matrix relating the complete set of steady state displacement responses to all possible sinusoidal force inputs. It is always symmetric for linear systems. It is the Fourier transform of the impulse response matrix.
DYNAMIC MODELLINGA modelling process where consideration as to time effects in addition to spatial effects are included. A dynamic model can be the same as a static model or it can differ significantly depending upon the nature of the problem.
DYNAMIC PRESSUREPressure due to local kinetic energy (1/2v2).
DYNAMIC SIMILARITYA similarity of forces.
DYNAMIC STIFFNESS MATRIXIf the structure is vibrating steadily at a frequency w then the dynamic stiffness matrix is (K+iwC-w2M).
DYNAMIC STRESSESStresses that vary with time and space.
DYNAMIC SUBSTRUCTURINGSpecial forms of substructuring used within a dynamic analysis. Dynamic substructuring is always approximate and causes some loss of accuracy in the dynamic solution.
EDDY BREAK UP MODELA reaction model in which the rate of reactant consumption is specified as a function of local flow turbulence properties and not the kinetic rate.
EDDY VISCOSITYA coefficient of proportionality between the Reynolds  stresses and the mean velocity gradients. Unlike the molecular viscosity, the turbulent viscosity is a property of the local state of the turbulence and not a property of the fluid. Its value varies from point to point in the fluid.
EFFECTIVE STRAIN (ALSO CALLED EQUIVALENT STRAIN)A scalar quantity defined (usually as the von Mises strain) to represent the individual strain components at any reference point also used for strain rates.
EFFECTIVE STRESS (ALSO CALLED EQUIVALENT STRESS)A scalar quantity defined (usually as the von Mises stress, q.v.) to represent the individual stress components at any reference point.
EIGENVALUE PROBLEMProblems that require calculation of eigenvalues and eigenvectors for their solution. Typically solving free vibration problems or finding buckling loads.
EIGENVALUESThe roots of the characteristic equation of a dynamic system. If the system has n equations of motion then it has n eigenvalues. The square root of the eigenvalues are the resonant frequencies. These are the frequencies that the structure will vibrate at if given some initial disturbance with no other forcing.
EIGENVALUES LATENT ROOTS CHARACTERISTIC VALUESThe roots of the characteristic equation of the system. If a system has n equations of motion then it has n eigenvalues. The square root of the eigenvalues are the resonant frequencies. These are the frequencies that the structure will vibrate at if given some initial disturbance with no other forcing. There are other problems that require the solution of the eigenvalue problem, the buckling loads of a structure are eigenvalues. Latent roots and characteristic values are synonyms for eigenvalues.
EIGENVECTORSThe displaced shape that corresponds to the eigenvalues. If the structure is excited at a resonant frequency then the shape that it adopts is the mode shape corresponding to the eigenvalue.
EIGENVECTORS LATENT VECTORS NORMAL MODESThe displacement shape that corresponds to the eigenvalues. If the structure is excited at a resonant frequency then the shape that it adopts is the mode shape corresponding to the eigenvalue. Latent vectors and normal modes are the same as eigenvectors.
EINSTEIN CONVENTIONIn tensor notation, whenever a certain index is repeated in the term, the term must be summed with respect to that index for all admissible values of the index. The summation convention allows us to omit writing the summation symbol.
ELASTIC FOLLOW-UPA structural phenomenon in which creep strain concentrates in rapidly creeping regions which are also relatively stiff; also analogously in plasticity.
ELASTIC FOUNDATIONIf a structure is sitting on a flexible foundation the supports are treated as a continuous elastic foundation. The elastic foundation can have a significant effect upon the structural response.
ELASTIC STIFFNESSIf the relationship between loads and displacements is linear then the problem is elastic. For a multi-degree of freedom system the forces and displacements are related by the elastic stiffness matrix.
ELASTIC UNLOADINGThis can occur in regions of structures that have become plastic and then have their stresses reduced to become elastic again, with plastic strains remaining.
ELASTIC-PLASTIC FRACTURE MECHANICSSee post yield fracture mechanics.
ELECTRIC FIELDSElectro-magnetic and electro-static problems form electric field problems.
ELEMENTIn the finite element method the geometry is divided up into elements. Each element has nodes associated with it. The behaviour of the element is defined in terms of the freedoms at the nodes.
ELEMENT ASSEMBLYIndividual element matrices have to be assembled into the complete stiffness matrix. This is basically a process of summing the element matrices. This summation has to be of the correct form. For the stiffness method the summation is based upon the fact that element displacements at common nodes must be the same.
ELEMENT STRAINS ELEMENT STRESSESStresses and strains within elements are usually defined at the Gauss points (ideally at the Barlow points) and the node points. The most accurate estimates are at the reduced Gauss points (more specifically the Barlow points). Stresses and strains are usually calculated here and extrapolated to the node points.
ELEMENT TYPESA formal definition of individual element formulations.
ELEMENTS (AND ELEMENT TYPES)The basic building blocks of the finite element model, which together form the (finite element) grid. In CFD, elements are normally triangular or quadrilateral in 2D and tetrahedra, prisms (wedges), pyramids or hexahedra in 3D.
ELLIPTIC EQUATIONSPartial differential equations of the form Auxx + 2Buxy + Cuyy = F(x,y,u,ux,uy) for which AC - B2 > 0. A number of classical steady-state mathematical descriptions of fluid flow and heat transfer are expressed as elliptic equations; examples include the irrotational flow of an incompressible fluid (the Laplace equation) and steady state conductive heat transfer. In a physical sense, elliptic equations describe behaviour in which the influence of a perturbation extends in all directions. For example, if the temperature is raised locally in a solid, heat is conducted away in all directions; or if a compressible fluid accelerates around an obstacle in steady irrotational flow, the effects of the acceleration are transmitted in all directions to the surrounding fluid. This behaviour contrasts with that described by parabolic and hyperbolic equations.
ENERGY DIFFERENCE TECHNIQUEUsed in fracture mechanics to evaluate the potential energy release rate at a single crack tip from the potential energies of two finite element runs differing only by a small change in crack length.
ENERGY EQUATIONAn equation derived from the first law of thermodynamics which states that the rate of change of energy of a fluid particle is equal to the rate of heat addition to the fluid particle plus the rate of work done on the particle.
ENERGY METHODS HAMILTONS PRINCIPLEMethods for defining equations of equilibrium and compatibility through consideration of possible variations of the energies of the system. The general form is Hamiltons principle and sub-sets of this are the principle of virtual work including the principle of virtual displacements (PVD) and the principle of virtual forces (PVF).
ENERGY RELEASE RATESee strain energy release rate.
ENGINEERING NORMALISATION MATHEMATICAL NORMALISATIONEach eigenvector (mode shape or normal mode) can be multiplied by an arbitrary constant and still satisfy the eigenvalue equation. Various methods of scaling the eigenvector are used Engineering normalisation - The vector is scaled so that the largest absolute value of any term in the eigenvector is unity. This is useful for inspecting printed tables of eigenvectors. Mathematical normalisation - The vector is scaled so that the diagonal modal mass matrix is the unit matrix. The diagonal modal stiffness matrix is the system eigenvalues. This is useful for response calculations.
ENGINEERING SIMULATIONThe use of numerical, physical or logical models of systems and scientific problems in predicting their response to different physical conditions.
ENGINEERING STRAIN (ALSO CALLED NOMINAL STRAIN)The ratio of the change in length over a given length to the original length.
EQUILIBRIUMThe state of a loaded body when the internal stresses are in equilibrium with the externally applied loads, and which always has to be achieved in finite element algorithms.
EQUILIBRIUM EQUATIONSInternal forces and external forces must balance. At the infinitesimal level the stresses and the body forces must balance. The equations of equilibrium define these force balance conditions.
EQUILIBRIUM FINITE ELEMENTSMost of the current finite elements used for structural analysis are defined by assuming displacement variations over the element. An alternative approach assumes the stress variation over the element. This leads to equilibrium finite elements.
EQUIVALENT MATERIAL PROPERTIESEquivalent material properties are defined where real material properties are smeared over the volume of the element. Typically, for composite materials the discrete fibre and matrix material properties are smeared to give average equivalent material properties.
EQUIVALENT STRAINSee effective strain.
EQUIVALENT STRESSSee effective stress.
ERGODIC PROCESSA random process where any one sample record has the same characteristics as any other record.
ERRORSSee accuracy, aliasing, convergence  error, diffusion error, discretisation  error, dispersive  error, dissipation  error, floating  point  errors, grid  independence, ill- posed  problem, modelling  errors, order  of  accuracy, residual, round-off  error, truncation error.
EULER EQUATIONSThe governing equations for inviscid compressible flow.
EULER-EULER MULTIPHASE METHODA multiphase method in which the different phases are treated as interpenetrating continua using the concept of a phasic volume fraction.
EULERIAN FORMULATIONA geometrically non-linear formulation where the equilibrium conditions are evaluated in the deformed configuration (q.v. also Lagrangian formulation).
EULERIAN FRAME OF REFERENCEA frame of reference based on a co-ordinate system as opposed to being based on a moving fluid element as used in the Lagrangian method.
EULERIAN METHOD LAGRANGIAN METHODFor non-linear large deflection problems the equations can be defined in various ways. If the material is flowing though a fixed grid the equations are defined in Eulerian coordinates. Here the volume of the element is constant but the mass in the element can change. If the grid moves with the body then the equations are defined in Lagrangian coordinates. Here the mass in the element is fixed but the volume changes.
EULER-LAGRANGE MULTIPHASE METHODA multiphase method in which the continuous phase is modelled using the Eulerian method and the dispersed phase (generally less than 15% volume fraction) is modelled using the Lagrangian method. See also Particle Source in Cell Method.
EXACT SOLUTIONSSolutions that satisfy the differential equations and the associated boundary conditions exactly. There are very few such solutions and they are for relatively simple geometries and loadings.
EXPANSION FACTORRatio of a dimension of adjacent grid cells.
EXPLICIT APPROACHA numerical scheme in which a single algebraic equation is used to evaluate each new nodal variable at a single time step.
EXPLICIT METHODS IMPLICIT METHODSThese are methods for integrating equations of motion. Explicit methods can deal with highly non-linear systems but need small steps. Implicit methods can deal with mildly non-linear problems but with large steps.
EXPLICIT SOLUTION SCHEMEAn algorithm, used in many time or load dependent analyses, whereby the solution for the next increment of time or load is obtained entirely from the solution and conditions at the previous step (q.v. also implicit solution). It is used in both static and dynamic analyses.
EXTERNAL FLOWSFlows over the external surface of an object (e.g. an aerofoil).
EXTRAPOLATION INTERPOLATIONThe process of estimating a value of a variable from a tabulated set of values. For interpolation values inside the table are estimated. For extrapolation values outside the table are estimated. Interpolation is generally accurate and extrapolation is only accurate for values slightly outside the table. It becomes very inaccurate for other cases.
FACETED GEOMETRYIf a curved line or surface is modelled by straight lines or flat surfaces then the modelling is said to produce a faceted geometry.
FAIL-SAFEA design philosophy in which products are designed in such a way that failures prior to the required operational life are not catastrophic.
FALSE DIFFUSIONSee numerical diffusion.
FAN MODELLINGEnables a geometric region to operate as a momentum and turbulence source to simulate the effect of a fan.
FAST FOURIER TRANSFORMA method for calculating Fourier transforms that is computationally very efficient.
FAVRE-AVERAGINGDensity weighted averaging used in deriving turbulent flow equations for cases where there are significant fluid density variations.
FCTSee flux-corrected transport method.
FDMSee finite difference method.
FEMSee finite element method.
FICK’S LAWStates that species diffuse in the direction of decreasing species concentration just as heat flows by conduction in the direction of decreasing temperature.
FIELD PROBLEMSProblems that can be defined by a set of partial differential equations are field problems. Any such problem can be solved approximately by the finite element method.
FINITE APPROXIMATIONSApproximations to a continuous function by representing it by finite quantities.
FINITE DIFFERENCE METHOD (FDM)A method for approximating gradients as part of the procedure for numerical solution of differential equations, by estimating a derivative by the ratio of two finite differences.
FINITE DIFFERENCE OPERATORSIdentify the type of differencing scheme applied e.g. forward, backward, central.
FINITE DIFFERENCESA numerical method for solving partial differential equations by expressing them in a difference form rather than an integral form. Finite difference methods are very similar to finite element methods and in some cases are identical.
FINITE ELEMENT METHOD (FEM)A computational method that originated from structural analysis but which is also applied to CFD, in which the computational  domain is subdivided into a finite number of elements over which discretised equations are solved.
FINITE VOLUME METHOD (FVM)A computational method in which the computational domain is subdivided into a finite number of control volumes over which discretised governing equations are solved. Primarily used for CFD.
FINITE VOLUME METHODSA technique related to the finite element method. The equations are integrated approximately using the weighted residual method, but a different form of weighting function is used from that in the finite element method. For the finite element method the Galerkin form of the weighted residual method is used.
FIRST ORDERAn approximation to an equation, or system of equations, where only the first terms in the Taylor expansions for functions are evaluated.
FIVE-POINT FORMULA FOR LAPLACE EQUATIONApproximation for solving the Laplace  equation by calculating derivatives using five discrete points in each co-ordinate direction.
FLAT PLATE FLOWFlow over a flat plate, has a well known boundary layer profile named after Blasius for incompressible flow and typically used as a validation test case.
FLEXIBILITY MATRIX FORCE METHODThe conventional form of the finite element treats the displacements as unknowns which leads to a stiffness matrix form. Alternative methods treating the stresses (internal forces) as unknowns leads to force methods with an associated flexibility matrix. The inverse of the stiffness matrix is the flexibility matrix.
FLOATING POINT ERRORSErrors which occur due to the representation of real numbers in digital computers by a finite number of digits or significant figures.
FLOW RULEUsed in plasticity to define a relationship between the plastic strain increment and the stress increment (q.v. also normality rule).
FLUID PROPERTIESThe collection of parameters that fully describe the physical properties of the fluid,e.g. density, viscosity, thermal conductivity.
FLUIDITY PARAMETERIn elastic-viscoplastic analysis, a parameter used in the evaluation of the viscoplastic strain rate.
FLUXAmount of transfer of fluid property (for example, enthalpy) through a specified surface or surface element.
FLUX DIFFERENCE SPLITTING SCHEMESA type of upwind discretisation scheme.
FLUX LIMITINGA technique for stabilising solution convergence, in the early stages of a solution, by limiting fluxes.
FLUX-CORRECTED TRANSPORT METHOD (FCT)A type of TVD (total variation diminishing) scheme which aims to correct the excessive dissipation of first order schemes without creating unwanted overshoots and oscillations, typical of second order schemes.
FLUX-VECTOR SPLITTING SCHEMESA type of upwind discretisation scheme.
FOLLOWER FORCESForces that change their direction to follow geometric deformation during a large deformation analysis.
FORCED RESPONSEThe dynamic motion resulting from a time varying forcing function.
FORCING FUNCTIONThe dynamic forces that are applied to a system.
FORCING FUNCTIONSThe dynamic forces that are applied to the system.
FORWARD DIFFERENCINGThe method by which the derivative of a variable at a point is approximated by the ratio of a) the difference in values of the variable at a forward point and the original point and b) the distance between the points.
FOURIER EXPANSIONS FOURIER SERIESFunctions that repeat themselves in a regular manner can be expanded in terms of a Fourier series.
FOURIER TRANSFORMA method for finding the frequency content of a time varying signal. If the signal is periodic it gives the same result as the Fourier series.
FOURIER TRANSFORM PAIRThe Fourier transform and its inverse which, together, allow the complete system to be transformed freely in either direction between the time domain and the frequency domain.
FRACTIONAL-STEP METHODA second order in time ADI (alternating direction implicit) method based on a factorisation of the Crank-Nicolson scheme method for deriving the pressure field, mostly used for LES and DNS.
FRACTURE PARAMETERS/CRITERIAThese are numerical quantities which represent the conditions at a crack tip in a given geometry at a given load level, e.g. , CTOD.
FRACTURE TOUGHNESS ( )For a given material, thickness and temperature, fracture toughness is the critical value of the stress intensity factor needed for a crack to grow under monotonic loading.
FRAMEWORK ANALYSISIf a structure is idealised as a series interconnected line elements then this forms a framework analysis model. If the connections between the line elements are pins then it is a pin-jointed framework analysis. If the joints are rigid then the lines must be beam elements.
FREE VIBRATIONThe dynamic motion that results from specified initial conditions. The forcing function is zero.
FREQUENCY DOMAINA structure’s forcing function and the consequent response is defined in terms of their frequency content. The inverse Fourier transform of the frequency domain gives the corresponding quantity in the time domain.
FRICTIONAL/FRICTIONLESS CONTACTIn contact analysis, the state of different surfaces coming into contact. Frictional is when the surfaces are sufficiently rough that friction is important and either sticking or slipping can occur. Frictionless is when the surfaces are assumed to be perfectly lubricated and so no friction occurs.
FRONTAL SOLUTION WAVEFRONT SOLUTIONA form of solving the finite element equations using Gauss elimination that is very efficient for the finite element form of equations.
FROUDE NUMBERA dimensionless quantity representing the ratio of inertia forces to gravitational forces, typically used in free-surface flows.
FULL APPROXIMATION SCHEMEMulti-grid technique for accelerating convergence rate in explicit solution methods.
FULL MULTIGRID METHODA method of increasing the speed of convergence of a solution by computing corrections on a coarser grid to remove low frequency components of errors, and transferring these corrections to the finer grid.
FULLY IMPLICITA method of solution whereby values are computed at all nodes simultaneously.
FVMSee finite volume method.
GALERKIN AND BUBNOV METHODThe Bubnov-Galerkin method is also known as the Galerkin  method where the weighting functions are made equal to the interpolation functions.
GALERKIN METHODA form of the method of weighted residuals. See weighted residual formulation.
GAP CHATTERINGThis occurs in contact analysis when certain gaps repeatedly open and close. This is an effect of the contact algorithm and can cause convergence problems.
GAP ELEMENTSSee contact elements.
GAUSS ELIMINATIONA systematic process of elimination for obtaining solutions to a set of linear equations.
GAUSS POINT EXTRAPOLATION GAUSS POINT STRESSESStresses calculated internally within the element at the Gauss integration points are called the Gauss point stresses. These stresses are usually more accurate at these points than the nodal points.
GAUSS POINTSStrategic locations within elements where numerical integration and stress evaluations are made. These vary over different element types and can differ depending on usage for both numerical integration and stress evaluation.
GAUSS THEOREMA theory which relates an integral throughout a volume to an integral over its bounding surface.
GAUSSIAN ELIMINATIONA form of solving a large set of simultaneous equations. Within most finite element systems a form of Gaussian elimination forms the basic solution process.
GAUSSIAN INTEGRATION GAUSSIAN QUADRATUREA form of numerically integrating functions that is especially efficient for integrating polynomials. The functions are evaluated at the Gauss points, multiplied by the Gauss weights and summed to give the integral.
GAUSSIAN QUADRATURESometimes known as Gaussian integration it is a commonly used form of evaluating numerically the integrals that appear in finite  element formulations. Generally, more sampling points (Gauss Points) in an element (see grid/mesh), where both the position and weighting is optimised, will reduce the integration error and give a more accurate solution.
GAUSS-SEIDEL ITERATION METHODAn iterative method of solving an equation of the form Ax=b, where A is a matrix and x & b are vectors, by iterating from an initial guess to the solution.
GENERALISED COORDINATESA set of linearly independent displacement coordinates which are consistent with the constraints and are just sufficient to describe any arbitrary configuration of the system. Generalised coordinates are usually patterns of displacements, typically the system eigenvectors.
GENERALISED CO-ORDINATESA system of non-orthogonal co-ordinates used for geometrical representation.
GENERALISED MASSThe mass associated with a generalised displacement.
GENERALISED STIFFNESSThe stiffness associated with a generalised displacement.
GEOGRAPHICAL NOTATIONA notation used in discretisation techniques for the values at and near to a node value (P) according to their relative position (N, n, S, s, E, e, W, w for north, south, east and west). f and b are sometimes used as analysis directions in the third dimension.
GEOMETRIC STIFFNESS STRESS STIFFNESSThe component of the stiffness matrix that arises from the rotation of the internal stresses in a large deflection problem. This stiffness is positive for tensile stresses and negative for compressive stresses. If the compressive stresses are sufficiently high then the structure will buckle when the geometric stiffness cancels the elastic stiffness.
GEOMETRICAL ERRORSErrors in the geometrical representation of the model. These generally arise from the approximations inherent in the finite element approximation.
GEOMETRICAL MODELThe representation of the physical geometry defining the shape and extent of the computational flow domain to be modelled.
GLOBAL CONSTRAINTA physical or numerical constraint that acts throughout the numerical model.
GLOBAL STIFFNESS MATRIXThe assembled stiffness matrix of the complete structure.
GODUNOV SCHEMEA method for discretising hyperbolic equations, which is often used in high speed, flow CFD codes.
GOVERNING EQUATIONSThe mathematical equations that describe the physics of the flow under consideration. These will typically be the conservation   equations of mass, momentum and energy but may additionally include equations for the transport of turbulence and species mass, for example.
GRAETZ NUMBERA dimensionless number representing the relative importance of conduction normal to the flow to thermal convection in the direction of the flow. It is the ratio of time required for heat conduction from the centre of a channel to the wall and the average residence time in the channel.
GRASHOF NUMBERThe fundamental dimensionless quantity for natural convection dominated flows. The Rayleigh number is often used in place of the Grashof number, being equal to the Grashof number multiplied by the Prandtl number.
GREEN’S STRAINA strain measure used in geometric non-linear analysis and defined, with reference to the original configuration, as the change in the squared length divided by twice the original squared length. It is given by (dS2 - dS02)/(2 dS02), where dS0 and dS are the undeformed and deformed lengths (see also Almansi strain).
GRIDA term used in CFD for the connecting lines between nodes, equivalent to a mesh in finite element methods.
GRID / MESHThe outcome of splitting up the computational  domain (discretisation) into a number of elements or cells defining the discrete points at which the numerical solution is computed. The points are normally the cell centres or cell vertices.
GRID ADAPTIONSee adaptive grid refinement.
GRID DENSITYThe number of cells in a given volume. A region of high grid density contains more cells than a region of low grid density. A higher grid density should be used in regions where the solution variables change rapidly so that their gradients can be computed and represented accurately. Lower grid density can be used where the solution is changing less in order to reduce the computational effort.
GRID GENERATIONThe act of generating a set of grid points for which the solution will be calculated.
GRID GROWTH RATEThe rate at which grid cell size changes from one cell to the next adjacent cell.
GRID INDEPENDENCEHaving run a simulation on a sequence of grids (usually refining each time) and found the same results for each grid, the solution is considered grid-independent. The converged solution is therefore independent of the size of grid (beyond a certain limit) used to obtain the solution.
GRID NON-UNIFORMITYA grid with varying grid density.
GRID POINTSThe discrete points that define the structure of the grid/mesh.
GRID REFINEMENTThe act of refining a grid such that the distance between adjacent grid points is reduced enabling a more accurate calculation and representation of the solution.
GRID REYNOLDS (PECLET) NUMBERAlso known as the cell Reynolds number or Peclet number.
GRID VELOCITYRepresents the velocity of the grid for problems involving grid movement.
GRIFFITH NUMBERA dimensionless quantity representing the relative importance of viscous dissipation to conduction. It is an indicator of the coupling of energy and momentum equations and is sometimes known as the Nahme number.
GROSS DEFORMATIONSDeformations sufficiently high to make it necessary to include their effect in the solution process. The problem requires a large deflection non-linear analysis.
GROSS YIELDINGIn elastic-plastic analysis, where widespread plasticity exists.
GUARD VECTORSThe subspace iteration (simultaneous vector iteration) method uses extra guard vectors in addition to the number of vectors requested by the user. These guard the desired vectors from being contaminated by the higher mode vectors and speed up convergence.
GUYAN REDUCTION METHODA method for reducing the number of degrees of freedom in a dynamic analysis. It is based upon a static approximation and always introduces some error in the computed dynamic solution. The error depends upon the choice of master freedoms.
GYROSCOPIC FORCESForces arising from Coriolis acceleration. These can destabilise a dynamic response and cause whirling.
HANGING NODESNodes not fully attached to all the surrounding elements. They can lead to an error in the finite element method where all nodes are assumed to be linked to elements.
HARDENINGIn non-linear material behaviour, the change in the current yield stress as plastic or creep straining occurs, such as work and strain hardening in plasticity, and time and strain hardening in creep.
HARDENING STRUCTUREA structure where the stiffness increases with load.
HARMONIC LOADINGA dynamic loading that is periodic and can be represented by a Fourier series.
HEAT CONDUCTIONThe analysis of the steady state heat flow within solids and fluids. The equilibrium balance between internal and external heat flows.
HEAT TRANSFERThe transfer of heat energy from one system to another. Heat transfer deals with the rate at which such energy is transferred.
HERMITIAN SHAPE FUNCTIONSShape functions that provide both variable and variable first derivative continuity (displacement and slope continuity in structural terms) across element boundaries.
HEXAHEDRAL ELEMENTSFinite elements with six faces, i.e. cuboid or brick elements.
H-GRIDA grid split open to form the shape of an H around a smooth body shape.
HIDDEN LINE REMOVALGraphical plots of models where non-visible mesh lines are not plotted.
HIERARCHICAL ELEMENTSElement families with varying shape function orders such that the stiffness matrix of each order contains the stiffness matrices of each of the lower ordered elements as sub-matrices.
HIGH ASPECT RATIO LOW ASPECT RATIOThe ratio of the longest side length of a body to the shortest is termed its aspect ratio. Generally bodies with high aspect ratios (long and thin) are more ill-conditioned for numerical solution than bodies with an aspect ratio of one.
HIGHER ORDERWhen a derivative of a partial differential equation is approximated, if the truncation error due to difference approximation is of the order two or more, then the difference scheme is known as a higher order scheme.
HOLONOMIC CONSTRAINTSConstraints that can be defined for any magnitude of displacement.
HOOKES LAWThe material property equations relating stress to strain for linear elasticity. They involve the material properties of Youngs modulus and Poisson ratio.
HOURGLASS EFFECTSSpurious element deformations due to zero energy modes (q.v.).
HOURGLASS MODEZero energy modes of low order quadrilateral and brick elements that arise from using reduced integration. These modes can propagate through the complete body.
H-REFINEMENT P-REFINEMENTMaking the mesh finer over parts or all of the body is termed h-refinement. Making the element order higher is termed p-refinement.
HYBRID COMPOSITEA composite with two or more types of reinforcing fibres.
HYBRID DISCRETISATION SCHEMEThe use of two or more different discretisation schemes depending upon some property of the flow, such as the Peclet number.
HYBRID ELEMENTSElements that use stress interpolation within their volume and displacement interpolation around their boundary.
HYBRID GRIDA computational grid containing more than one cell or element type.
HYDROSTATIC (ALSO CALLED VOLUMETRIC) STRESS AND STRAINIs the average of the direct stress or strain components at any point of reference, ignoring the shear components. It causes change in volume but not change in shape of an element of material (q.v. also deviatoric stress and strain).
HYDROSTATIC PRESSUREThe pressure due to depth, calculated by the product of density, gravity and depth.
HYDROSTATIC STRESSThe stress arising from a uniform pressure load on a cube of material. It is the average value of the direct stress components at any point in the body.
HYPERBOLIC EQUATIONSPartial 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 hyperbolic equations are steady inviscid two-dimensional supersonic flow, and time dependant problems with negligible dissipation such as the wave equation. Hyperbolic equations also dominate the analysis of vibration problems. An important feature of phenomena governed by hyperbolic equations, is that there exists from any point in the mathematical space in which the equations operate, a set of "characteristics"- lines (or surfaces in 3-D) along which the partial differential equations can be reduced to ordinary differential equations. This feature allows the use of special and very efficient computational algorithms to solve the equations, based on the "Method of Characteristics".
HYPERELASTICITYA material which possesses an elastic potential function, known as the strain energy function, which is a scalar function of strain and whose derivatives with respect to each strain component gives the corresponding stress component.
HYSTERETIC DAMPINGA model for the dissipation of energy in which the damping force is proportional to the amplitude of the displacement and opposes the velocity of motion.
IGES (INTERNATIONAL GRAPHICS EXCHANGE STANDARD)A neutral file format used to translate geometrical information between different CAD, CAE and analysis software packages.
ILL-CONDITIONING ERRORSNumerical (rounding) errors that arise when using ill-conditioned equations.
ILL-CONDITIONING ILL-CONDITIONED EQUATIONSEquations that are sensitive to rounding errors in a numerical operation. The numerical operation must also be defined. Equations can be ill-conditioned for solving simultaneous equations but not for finding eigenvalues.
ILL-POSED PROBLEMA problem in which the description of the problem is not self consistent, is not complete or is overconstrained.
IMPLICIT APPROACHA numerical scheme in which the solution of the entire grid is required for each time level. For a single time level, it is very computationally expensive compared to the explicit approach but can often be used with much larger intervals between time levels (i.e. much larger time steps).
IMPLICIT SOLUTION SCHEMEAn algorithm, usable in many types of non-linearity, whereby the solution for the next increment of time or load is obtained from the solution at the previous step and conditions from the current step (q.v. also explicit solution). It is used in both static and dynamic analyses.
IMPULSE RESPONSE FUNCTIONThe response of a system to an applied impulse.
IMPULSE RESPONSE MATRIXThe matrix of all system responses to all possible impulses. It is always symmetric for linear systems. It is the inverse Fourier transform of the dynamic flexibility matrix.
INCOMPRESSIBILITYStraining with zero volumetric strain (i.e. no change in volume).
INCOMPRESSIBLE FLOWFlow where the density is not a function of pressure, i.e. the flow remains at a constant density (less than approximately Mach 0.3) in all locations.
INCREMENTAL FORMULATIONThe splitting up of applied load or time into small quantities (increments or steps) such that within each a meaningful converged solution can be conducted.
INCREMENTAL SOLUTIONA solutions process that involves applying the loading in small increments and finding the equilibrium conditions at the end of each step. Such solutions are generally used for solving non-linear problems.
INDIRECT METHODSee iterative method.
INELASTIC MATERIAL BEHAVIOURA material behaviour where residual stresses or strains can remain in the body after a loading cycle, typically plasticity and creep.
INERTANCE (ALSO CALLED ACCELERANCE)The ratio of the steady state acceleration response to the value of the forcing function.
INERTIA FORCEThe force that is equal to the mass times the acceleration.
INITIAL BOUNDARY VALUE PROBLEMA problem for which the solution can be obtained by specifying two initial conditions and a boundary  condition. Hyperbolic  problems are initial boundary value problems.
INITIAL BUCKLINGThe load at which a structure first buckles.
INITIAL CONDITIONSConditions at the initial (start) time in a time dependant simulation.
INITIAL STIFFNESS METHODA modified Newton-Raphson solution in which the initial linearly elastic, small displacement, stiffness matrix is used throughout the analysis.
INITIAL STRAINSThe components of the strains that are non-elastic. Typically thermal strain and plastic strain.
INNER ITERATIONSAn iterative step embedded within an iterative scheme. For example, a single iterative step or iteration may require the solution to a set of equations. These equations may themselves be solved by an iterative method. The iterations required to determine the solution to this set of equations can be considered to be “inner iterations” within the overall iterative scheme.
INTEGRATION BY PARTSA method of integrating a function where high order derivative terms are partially integrated to reduce their order.
INTERFACE CAPTURING METHODA method for identifying interfaces caused by severe density or other property changes.
INTERFACE TRACKING METHODA numerical approach for tracking interfaces (see Interface Capturing).
INTERNAL FLOWSA fluid flow domain that is contained by and passes through a solid structure. All boundaries of the domain can be defined as walls, periodic  boundaries, inlets or outlets. Compare with external flows.
INTERPOLATION FUNCTIONS SHAPE FUNCTIONSThe polynomial functions used to define the form of interpolation within an element. When these are expressed as interpolations associated with each node they become the element shape functions.
INVISCID FLOWFlows for which viscosity or shear effects can be neglected. By making this assumption the Euler  equations (a subset of the Navier-Stokes  equations) can be used, simplifying the solution techniques.
IRREGULAR GRIDSometimes known as an unstructured  grid (although a regular grid can also be unstructured). An irregular grid has no regular array of cells that can be grouped into rows, columns and layers.
IRROTATIONAL FLOWFlows in which the curl of the velocity is equal to zero. In physical terms, individual elements of fluid have motion described by translation without rotation. By making this assumption along with the inviscid assumption a potential  flow problem can be solved.
ISOPARAMETRIC ELEMENTSElements in which the displacements and geometry variation within an element are represented by the same shape functions (q.v.).
ISOTROPIC HARDENINGThis occurs when, as plastic strains increase after initial yielding, the yield surface in principal stress coordinates expands uniformly about the origin while still maintaining its shape and orientation.
ISOTROPIC MATERIALMaterials where the material properties are independent of the co-ordinate system.
ITERATIONA step in an iterative process. See iterative method.
ITERATIVE METHODA process in which the equations are not solved directly but indirectly by a series of iterative steps or iterations. An initial estimate of the solution is made, and an algorithm defined whereby the estimate is improved until it satisfies the equations to within some specified tolerance (see convergence criterion). Linear systems can be solved directly in one step by direct methods. Non-linear systems, typical of CFD problems, will necessarily be iterative.