NAFEMS Glossary Terms A-C
ACCELERATIONThe second time derivative of the displacement (the first time derivative of the velocity).
ACCURACYA measure of the similarity of a simulation to the physical flow it is intended to represent. See also model accuracy, numerical accuracy, validation and verification.
ADAMS METHODSA common type of multi-point temporal scheme that requires several time levels as opposed to the usual two. It can be generated by fitting a polynomial to variables through time.
ADAMS-BASHFORTH METHODExplicit Adams method.
ADAMS-MOULTONImplicit Adams method.
ADAPTIVE GRID REFINEMENTRefinement of a computational grid based, for example, on regions with steep flow gradients. This can be an automatic, manual or semi-manual procedure.
ADAPTIVITYSee mesh adaptivity.
ADDITIVE DECOMPOSITIONDecomposition of an array [A] into components of which the original matrix is the sum, i.e. [A] = [B] + [C]
ADI TECHNIQUEThe Alternating-Direction-Implicit technique, which is generally a temporal solution approach, where the use of explicit and implicit solution techniques is alternated with time in different co-ordinate directions.
ADIABATIC WALL CONDITIONA perfectly thermally insulated or lagged wall, represented by a boundary condition of zero normal heat flux.
AEROELASTICITYFluid / structure interaction between elastic components (e.g. wings and aerofoils) and the surrounding fluid flow field. Also occurs in turbomachinery and heat exchangers.
ALGEBRAIC EIGENVALUE PROBLEMThe eigenvalue problem when written in the form of stiffness times mode shape minus eigenvalue times mass times mode shape is equal to zero. It is the form that arises naturally from a discrete parameter model in free vibration.
ALGEBRAIC GRID GENERATIONA grid generation method in which the mesh is interpolated from the bounding, geometry-constrained edges. See also differential grid generation.
ALGEBRAIC MULTIGRIDA technique for speeding up the solution of an iterative technique by reducing the number of iterations necessary for convergence. It involves the systematic coarsening of the original computational grid into a series of coarser grids. In addition to solving the equations for the original grid, equivalent equations are also solved on each coarser grid, transferring corrections onto the finer levels. This allows the solution procedure to take into account the overall solution error and global continuity, thus reducing long wavelength errors.
ALGEBRAIC STRESS MODEL (ASM)A type of turbulence model that solves for the Reynolds stresses but ignores transport terms. The model is a simplification of a full differential “RSM” (Reynolds Stress Model).
ALMANSI STRAINStrain defined in the deformed state as changes in squared length per twice the new squared length. It is given by (dS2 - dS02)/(2 dS2), where dS0 and dS are the undeformed and deformed lengths (see also Green’s strain).
ALTERNATING PLASTICITYOccurs in cyclic loading (q.v.) when there is a progressive increase in total strain with each cycle.
ALTERNATING-DIRECTION-IMPLICIT TECHNIQUESee ADI.
AMDAHL’S LAWWhen parallel processing at the loop level (using, for example, an auto-parallelising compiler) the least efficient part of the code (that which would not parallelise) will strongly limit the potential increase in computational performance.
AMPLIFICATION FACTORA concept arising from Von Neumann stability analysis. If en+1 is an error at a time level n+1 and en a value at a time level n the amplification factor is defined as A = en+1 /en. For a stable scheme A 1 , i.e. the error decreases with time.
AMPLIFICATION MATRIXAmplification factors are derived considering a single equation. Typical Computational fluid dynamics (CFD) problems involve sets of equations in matrix form. An amplification matrix extends the amplification factor concept to groups of equations.
ANALYTICAL DOMAINSee domain.
ANALYTICAL SOLUTIONA solution that is obtained directly using analytical methods as opposed to using computational or iterative methods.
ANISOTROPYA material where the response to load depends on the direction within the material. In general, 21 independent constants are required to relate stress and strain.
APPROXIMATE FACTORISATION TECHNIQUEA manipulation (factorisation) of equations to produce a more convenient or efficient solution form, without sacrificing the formal order of the numerical scheme. Also known as splitting.
ARBITRARY LAGRANGIAN EULERIAN MESH UPDATINGAn automatic mesh re-zoning algorithm where a manual decision is replaced by regular re-zoning operations carried out at fixed time increments or number of calculation cycles.
ARC LENGTH METHODA non-linear iterative technique used to solve non-linear problems at or near limit points, where there is a change in sign of the slope of the load-displacement curve.
AREA COORDINATESA special coordinate system that is used for defining shape functions for triangular and tetrahedral elements.
ARRHENIUS KINETIC RATEAn expression to determine the reaction rate in a kinetically limited reaction.
ARTIFICIAL COMPRESSIBILITYSome CFD methods for compressible flow combine the continuity equation with the equation of state to yield an equation for pressure. These methods are extended to incompressible flow by adopting an ‘equation of state’ for the fluid containing a small amount of compressibility.
ARTIFICIAL DISSIPATIONSee numerical dissipation.
ARTIFICIAL VISCOSITYAn inaccuracy arising from the discretisation process that manifests itself as an apparent increase in the specified fluid viscosity. Artificial viscosity improves the stability of a solution at the expense of solution accuracy.
ASMSee Algebraic Stress Model.
ASPECT RATIOA measure of quality for a computational grid. In two-dimensions, the ratio of cell height to cell width.
ASPECT RATIOSThe ratio of the different element side or edge lengths, used for establishing amounts of distortion (q.v.).
ASSEMBLYThe process of assembling the element matrices together to form the global matrix. Typically element stiffness matrices are assembled to form the complete stiffness matrix of the structure.
ASSOCIATIVE PLASTICITYA form of plasticity in which the yield function and the plastic potential are identical.
A-STABILITYAn unconditionally stable temporal scheme (in practice it corresponds to an implicit temporal scheme).
AUGMENTED LAGRANGIAN METHODA combination of the penalty function and Lagrange multiplier methods (q.v.). Used in contact analysis, where the contact force is defined in terms of the Lagrange multiplier plus a penalty stiffness term.
AUTOMATIC LOAD/TIME INCREMENTATIONA method for automatic incrementation in an applied load or time incremental-iterative solution process, i.e. the increment sizes are not specified by the analyst.
AUTOMATIC MESH GENERATIONThe process of generating a mesh of elements over the volume that is being analysed. There are two forms of automatic mesh generation: Free Meshing - Where the mesh has no structure to it. Free meshing generally uses triangular and tetrahedral elements. Mapped Meshing - Where large regions, if not all, of the volume is covered with regular meshes. This can use any form of element. Free meshing can be used to fill any shape. Mapped meshing can only be used on some shapes without elements being excessively distorted.
AUTOMATIC NODE RENUMBERING BANDWITH PROFILE WAVEFRONTThe process of renumbering the nodes or elements to minimise the bandwidth, the profile or the wavefront of the assembled matrix. This renumbering is normally transparent to the user.
AXIS BOUNDARY CONDITIONA boundary condition at the centreline of an axi-symmetric geometry.
AXISYMMETRIC ELEMENTAn element defined by rotating a cross-section about a centre line.
AXI-SYMMETRIC GRIDA cylindrical polar co-ordinate system grid in which all derivatives with respect to the tangential co-ordinate direction are assumed to be zero.
AXISYMMETRIC THIN SHELL AXISYMMETRICAL THICK SHELLAn element forms an axisymmetric thin shell if a line element is rotated about an axis. An element forms an axisymmetric thick shell if a triangular or quadrilateral element is rotated about an axis.
AXISYMMETRYIf a shape can be defined by rotating a cross-section about a line (e.g. a cone) then it is said to be axisymmetric. This can be used to simplify the analysis of the system. Such models are sometimes called two and a half dimensional since a 2D cross-section represents a 3D body.
BACK SUBSTITUTIONThe final phase when solving simultaneous equations using Gaussian elimination.
BACKSCATTERUsually, turbulence energy is dissipated from larger to smaller eddies. The reverse can also occur and this is called backscatter.
BACKWARD 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 backward point and the original point and b) the distance between the points.
BACKWARD STAGGERED GRIDSee staggered grid.
BACKWARDS FACING STEPThis is a standard CFD benchmark test. It is a channel flow where the lower channel wall is constructed to produce a sudden expansion (a step change) in the channel height. The step change forces separation and produces a simple test case for evaluating the performance of algorithms when modelling separated flows.
BANDED MATRIXA matrix that has a structured appearance, the elements appearing as neat adjacent diagonal lines.
BANDWIDTHThe half bandwidth of a matrix is the maximum distance of any non-zero term in the matrix from the leading diagonal of the matrix. The bandwith for a symmetric matrix is then twice this.
BARLOW POINTSThe set of Gauss integration points that give the best estimates of the stress for an element. For triangles and tetrahedra these are the full Gauss integration points. For quadrilateral and brick elements they are the reduced Gauss points.
BASIS FUNCTIONSAlso known as shape or interpolation functions, are used to calculate the value of a variable over an element in terms of the discrete values at the nodes.
BASIS SPACEWhen an element is being constructed it is derived from a simple regular shape in non-dimensional coordinates. The coordinates used to define the simple shape form the basis space. In its basis space a general quadrilateral is a 2x2 square and a general triangle is an isosoles triangle with unit side lengths.
BAUSCHINGER EFFECTObserved in plasticity when, after initial tensile loading into the plastic region, the yield stress in compression is less than the equivalent value in tension.
BEAM AND WARMING SCHEMEA modification to the Crank-Nicolson scheme that improves its speed of convergence. The modification treats terms in which the transported variable and variable transporting it are the same i.e. terms of the form uu / x .
BEAM ELEMENTA line element that has both translational and rotational degrees of freedom. It represents both membrane and bending actions.
BEM (BOUNDARY ELEMENT METHOD)A numerical solution method where only the boundary of a domain is discretised with elements. There are no elements covering the interior domain. A limitation of the method is that only problems with constant interior properties can be solved.
BENCHMARKINGThe process of performance testing relative to some performance indicator (a benchmark).
BENDINGBending behaviour is where the strains vary linearly from the centre line of a beam or centre surface of a plate or shell. There is zero strain on the centre line for pure bending. Plane sections are assumed to remain plane. If the stresses are constant normal to the centre line then this is called membrane behaviour.
BERNOULLI EQUATIONThis refers to an equation that expresses conservation of fluid kinetic energy, gravitational potential energy and energy associated with pressure in the absence of all other energy transfer mechanisms, including viscous dissipation.
BICONJUGATE GRADIENT METHODAn iterative method for solving large systems of algebraic equations. The solution is obtained by aiming for the minimum residual by choosing next search direction vectors and bi-directional vectors which are, as nearly as possible, in the directions of steepest descent. These directions are subject to the overriding condition that they are orthogonal with respect to the coefficient matrix.
BIFURCATIONOccurs on a non-linear load-displacement curve as the load path forks into two or more solution paths that satisfy equilibrium. Only one path is stable, the others being unstable.
BLOCK STRUCTURED GRIDA grid that comprises of several connected structured sub-grids (blocks).
BODY FITTED CO-ORDINATESThe use of a co-ordinate system fitted to the geometry such that the grid points lie on the domain surfaces. Such grids frequently have the accuracy advantage that the grid lines are approximately parallel or orthogonal to flow streamlines.
BODY FORCEA force acting on the fluid in the frame of reference of the calculation due to effects other than pressure and viscosity, e.g. gravitational or centripetal forces, magnetic or electrostatic fields or general motion of the frame of reference.
BODY FORCE VECTORMechanical loadings within the interior of the volume, typically inertia loadings in a stiffness analysis.
BOUNDARY CONDITIONSpatial or temporal specification of variable values or behaviour necessary to produce a unique solution.
BOUNDARY CONDITIONSPrescribed degrees of freedom and other quantities within a finite element model, which represent the physical model and are required to produce a unique solution for any type of applied loading.
BOUNDARY ELEMENT BOUNDARY INTEGRALA method of solving differential equations by taking exact solutions to the field equations loaded by a point source and then finding the strengths of sources distributed around the boundary of the body required to satisfy the boundary conditions on the body.
BOUNDARY ELEMENT METHODSee BEM.
BOUNDARY FITTED CO-ORDINATESSimilar to body fitted co-ordinates.
BOUNDARY LAYERThe layer of fluid adjacent to solid surfaces that has been affected, through viscous action, by the presence of the solid surface. There are many mathematical descriptions for the boundary layer’s extent and these are required in many turbulence models. The simplest description is the region, adjacent to a solid surface, where the fluid velocity is less than the free stream velocity (velocity outside the boundary layer) by more than 1 %.
BOUNDARY POINTSPoints on the boundary of a domain.
BOUNDARY VALUE PROBLEMA problem where the final solution is dependent on just the boundary conditions and not the initial conditions.
BOUNDEDNESSA property of a numerical scheme in which the predicted values are limited within certain physically realistic bounds.
BOUSSINESQ APPROXIMATIONIn CFD there are two types of Boussinesq approximation: In purely buoyancy driven flows, where density variations are small, it can be possible to ignore density variations in all equations except the source term for the velocity component equation that is parallel to the gravity vector. In turbulent flows, it is widely used to approximate the relationship between Reynolds stresses and eddy viscosity (multiplied by the fluid mean strain rate).
BOW SHOCK WAVEShock wave occurring at the bow or leading edge of an object.
BRILEY AND MCDONALD METHODA lesser used alternative to the Beam Warming Scheme.
BRITTLE FRACTUREThis is the type of fracture occurring for a crack in a material whose behaviour is described as brittle, when any plastic deformation is very limited so that fracturing occurs without significant prior deformation. This is typified by glassy materials and metals at temperatures below the range of the brittle-ductile transition temperature.
BUBBLE FUNCTIONSElement shape functions that are zero along the edges of the element. They are non-zero within the interior of the element.
BUBNOV-GALERKIN METHODA way of referring to the standard Galerkin Finite Element method where, in the discretisation process, weighting functions are equal to shape functions.
BUCKLINGBuckling is a geometric instability, generally caused by compressive forces in thin-sectioned bodies. It can be analysed as a special case of geometric non-linearity using eigenvalue analysis.
BUCKLING (SNAP THROUGH)The situation where the elastic stiffness of the structure is cancelled by the effects of compressive stress within the structure. If the effect of this causes the structure to suddenly displace a large amount in a direction normal to the load direction then it is classical bifurcation buckling. If there is a sudden large movement in the direction of the loading it is snap through buckling.
BUFFER LAYERA region in the turbulent boundary layer linking the viscous sub-layer to the fully turbulent zone.
BUOYANCY TERMA body force term associated with density changes. These can be due to temperature differences or changes in species concentrations.
BURGER’S EQUATIONA non-linear, one-dimensional idealised form of the Euler equation (the Navier- Stokes equation with zero viscosity). It is often used for the detailed mathematical analysis of solution procedures.
CALORIFICALLY PERFECT GASA gas for which a linear relationship exists between temperature and internal energy. See thermally perfect gas and perfect gas.
CAM-CLAY MODELA model describing the behaviour of clay-type soils, using a hardening/softening elastic-plastic constitutive law based on the critical state framework whose yield surface plots as a logarithmic curve
CAPILLARY CONVECTIONWhen a free surface has a significant temperature gradient, the variations in surface tension force (which is a function of temperature) can cause a fluid shear stress to arise. The fluid tends to move from the region of high to low temperature and this process is called capillary or Marangoni convection.
CARTESIAN GRIDA grid in which lines of constant x, constant y and constant z are orthogonal.
CAUCHY STRESSSee true stress.
CCCT (CURVATURE COMPENSATED CONVECTIVE TRANSPORT)A convective term treatment (i.e. a means of interpolating to control volume faces from adjacent grid points).
CEBECI-SMITH METHODA density weighted technique for defining mean turbulent variables that reduces the number of products of density fluctuations with other fluctuating quantities (these authors also devised a popular mixing length turbulence model).
CELLDiscrete area or volume over which governing equations are integrated. The complete group of cells should define the domain under consideration
CELL REYNOLDS NUMBERSee Peclet number.
CELL-CENTRED SCHEMEA discretisation scheme in which values of the dependent variables are stored at the centre of each cell.
CELL-VERTEX SCHEMEA discretisation scheme in which values of the dependent variables are stored at the vertices of each cell.
CENTRAL COEFFICIENTCoefficient associated with the node at the centre of a cell.
CENTRAL DIFFERENCE METHODA method for numerically integrating second order dynamic equations of motion. It is widely used as a technique for solving non-linear dynamic problems.
CENTRAL DIFFERENCING SCHEMEA discretisation approach in which the convective terms are calculated using a polynomial representation for the quantity of interest, with the polynomial centred on the point of interest. It may be a first order method where a simple linear average is used but is more often referred to as a second order method where the solution is represented as a quadratic. It may also be a higher order method. See discretisation schemes.
CFDSee Computational Fluid Dynamics.
CFL CONDITIONThe Courant-Friedrich-Lewy condition states that the Courant number should be less than or equal to unity.
CGM (CONJUGATE GRADIENT METHOD)A method for solving non-linear simultaneous equation sets that involves searching for the minimum of a function.
C-GRIDA curvilinear grid that is wrapped round an object in a ‘C’ shaped form.
CHARACTERISTIC LINESLines along which the derivatives of the velocity components are indeterminate and across which they may be discontinuous.
CHARACTERISTIC VALUESame as the eigenvalue.
CHARACTERISTIC VECTORSame as the eigenvector.
CHEBYSHEV ACCELERATIONA technique for accelerating convergence of the crude Jacobi method (a method for solving simultaneous equation sets).
CHEBYSHEV POLYNOMIALAn orthogonal function that can be used in spectral type methods. Cosine based expression can also be used to generate meshes suitable for resolving laminar boundary layers (see NAFEMS document on CFD meshes)
CHECKER-BOARD PRESSURE FIELDA pressure field of alternating values, in the manner of a chessboard. It is obtained by using a solution technique that wrongly ignores the influence of every other pressure node in the solution procedure. This problem is overcome by using staggered grids or special ‘momentum interpolation’ techniques such as that proposed by Rhie and Chow.
CHIMERA GRIDA Chimera grid comprises sub-grids of different natures that overlay at edges and enable the mapping of complex geometries. The method is well suited to the modelling of moving bodies. A mesh of a particular type can be wrapped round the body and this mesh can move through a background mesh that conforms to the main fluid region.
CHOLESKI FACTORISATIONA technique used for the decomposition of a matrix into upper and lower triangles. It is used in the application of the basic Gaussian elimination procedure and is suitable only for the solution of positive definite systems in which all the eigenvalues of the matrix are positive.
CHOLESKY FACTORISATION (SKYLINE)A method of solving a set of simultaneous equations that is especially well suited to the finite element method. It is sometimes called a skyline solution. Choose to optimise the profile of the matrix if a renumbering scheme is used.
CLEBSCH REPRESENTATIONAn economical representation (in terms of the number of solution variables) for inviscid rotational flows. For practical cases, it is generally restricted to steady flows.
CLOSED-FORM DISPLACEMENT METHODFor fracture mechanics, a special form of displacement substitution that only uses the calculated values in the crack tip elements.
CLOSUREGenerally used in relation to turbulence modelling. For turbulent flows, the governing equations (when the RANS approach is used) have turbulence correlations that need to be accounted for using empirically based models. Such models enable closure of the problem i.e. give sufficient equations for the number of unknowns, thus enabling a solution to be produced. Examples of closure models include k-, RSM etc..
COEFFICIENT OF VISCOUS DAMPINGThe constants of proportionality relating the velocities to the forces.
COINCIDENT NODESNodes that occupy the same location in space and that may result in collapsed cells or grid discontinuities.
COLE-HOPF TRANSFORMATIONA mathematical transformation that allows the analytical solution of Burger’s equation for many combinations of initial and boundary conditions.
COLLAPSED ELEMENTAn element in which two or more nodes are coincident, sometimes known as a degenerate element.
COLLOCATED GRIDA computational grid in which collocation of solution variables is applied.
COLLOCATIONIn most modern CFD codes, the variables are all located in the same place (either a cell centre, cell vertex or cell face centre). However, in the past, problems coupling the velocity and pressure fields resulted in variables being stored in different locations. This approach has some computational advantages but does not lend itself to complex general geometry solution procedures and efficient coding.
COLUMN VECTOR (COLUMN MATRIX)An nx1 matrix written as a vertical string of numbers. It is the transpose of a row vector.
COMPACT DIFFERENCINGA differencing scheme which uses close neighbours to obtain differencing methods that have an accuracy greater than second order.
COMPATIBILITY EQUATIONSCompatibility is satisfied if a field variable, typically the structural displacement, which is continuous before loading is continuous after loading. For linear problems the equations of compatibility must be satisfied. Nonlinearity in or non-satisfaction of, the compatibility equations leads to cracks and gaps in the structure. For finite element solutions compatibility of displacement is maintained within the element and across element boundaries for the most reliable forms of solution.
COMPATIBILITY OF STRAINSCompatibility of strain is satisfied if strains that are continuous before loading are continuous after. Admin
COMPLETE DISPLACEMENT FIELDWhen the functions interpolating the field variable (typically the displacements) form a complete n'th order polynomial in all directions.
COMPLETENESSA property of an iterative method in which the approximate solution converges to the exact solution.
COMPLEX EIGENVALUESThe eigenvalues of any damped system. If the damping is less than critical they will occur as complex conjugate pairs even for proportionally damped systems. The real part of the complex eigenvalue is a measure of the damping in the mode and should always be negative. The imaginary part is a measure of the resonant frequency.
COMPLEX EIGENVECTORSThe eigenvectors of any damped system. For proportionally damped systems, they are the same as the undamped eigenvectors. For non-proportionally damped systems with damping in all modes less than critical they are complex numbers and occur as complex conjugate pairs.
COMPOSITE GRIDSComplex geometries are sometimes modelled using several relatively simple connected or ‘Composite Grids’. This is sometimes called the multiblock approach.
COMPOSITE MATERIALA material that is made up of discrete components, typically a carbon-epoxy composite material or a glass-fibre material. Layered material and foam materials are also forms of composite materials.
COMPRESSIBLE FLOWFlow (of gases) where speeds are sufficiently high, causing significant fluid density changes. In some cases (where the Mach number exceeds unity) pressure discontinuities, known as shocks, may occur. A commonly used ‘rule of thumb’ for judging whether a flow is compressible is if the Mach number exceeds 0.3 in one or more regions.
COMPUTATIONAL DOMAINSee domain.
COMPUTATIONAL EFFICIENCYA general phrase that refers to how economical a computer program is with respect to storage or processing power.
COMPUTATIONAL FLUID DYNAMICS (CFD)The field of solving complex non-linear differential equations governing fluid flow using computer.
COMPUTATIONAL MOLECULEIn CFD, variables are considered to be stored in different discrete points in both space and time. The computational molecule uses lines to show the connectivity and topology of nodes and / or cells associated with the discretisation process for a single solution point.
COMPUTATIONAL PLANEAn approach for modelling flows in complex geometries that involves the transformation of the governing equations for a simple co-ordinate system (say x-y) into a co-ordinate system that matches the shape of the geometry (see Conformal Co-ordinates). The governing equations for the co-ordinate system that matches the complex geometry are generally more complex than the original equations and are solved in what is called the computational plane. The grid in the computational plane has a uniform orthogonal form and hence requires less sophisticated solver technology.
CONDENSATION STATIC CONDENSATION MODAL CONDENSATIONThe reduction of the size of a problem by eliminating (condensing out) some degrees of freedom. For static condensation the elimination process is based upon static considerations alone. In more general condensation it can include other effects, typically model condensation includes both static and dynamic effects.
CONDITION NUMBER (CFD)The ratio of the maximum to the minimum eigenvalues of a matrix. Condition number values much larger than one can lead to very slow convergence of a CFD problem. To overcome, this preconditioning can be used.
CONDITION NUMBER (FEA)The ratio of the highest eigenvalue to the lowest eigenvalue of a matrix. The exponent of this number gives a measure of the number of digits required in the computation to maintain numerical accuracy. The higher the condition number the more chance of numerical error and the slower the rate of convergence for iterative solutions.
CONDITIONAL STABILITYStability that is conditional on some criteria being fulfilled (see stability criterion and the CFL condition).
CONDITIONAL STABILITY UNCONDITIONAL STABILITYAny scheme for numerically integrating dynamic equations of motion in a step by step form is conditionally stable if there is a maximum timestep value that can be used. It is unconditionally stable (but not necessarily accurate) if any length of time step can be used.
CONDUCTIONA mode of heat transfer in which the heat energy is transferred on a molecular scale with no movement of macroscopic particles (matter) relative to one another: described by Fourier’s law.
CONFORMAL CO-ORDINATESCo-ordinates that conform to the shape of a generally fairly complex region. For example, when modelling an aerofoil a co-ordinate system with lines that wrap round the wing could be used.
CONFORMAL MAPPINGThe use of mathematical transformations to solve equations for relatively complex geometries. Mapping enables equations to be solved on a relatively simple domain.
CONFORMING ELEMENTAn element in which inter-element continuity conditions are satisfied along the complete extent of inter-element boundaries.
CONGRUENT TRANSFORMATIONA transformation of the coordinate system of the problem that preserves the symmetry of the system matrices.
CONJUGATE GRADIENT METHODA method for solving simultaneous equations iteratively. It is closely related to the Lanczos method for finding the first few eigenvalues and eigenvectors of a set of equations.
CONJUGATE GRADIENT METHOD (CGM)See CGM.
CONNECTIVITYWhen using unstructured grids, it is necessary to express which Node are connected to each other. These data are called connectivity information and are usually stored in look-up tables.
CONSERVATIONThe preservation of an extensive property in a closed system, see conservative variables
CONSERVATION FORM OF EQUATIONSEquations written in a form that directly represents the quantity conserved; mass, momentum, energy, rather than velocity and temperature. The equations can then be expressed as: Rate of Change of Conserved Quantity = Diffusion + Convection + Sources - Sinks
CONSERVATION OF ENERGYThe energy entering or leaving a volume of fluid due to flow convection and conduction is balanced by the energy of the fluid volume over time and the dissipation due to viscous forces.
CONSERVATION OF MASSThe condition that mass cannot be created or destroyed within a fluid flow system.
CONSERVATION OF MOMENTUMThe condition that the forces on a fluid in a certain volume equal the mass of that fluid multiplied by its acceleration, effectively Newton’s second law of motion.
CONSERVATIVE DISCRETISATION SCHEMEA numerical scheme in which the discretisation of the algebraic equation describing the transport processes for a dependant variable is such that conservation of the associated extensive property is mathematically assured.
CONSERVATIVE FORM OF FLOW EQUATIONSAn equation form that, regardless of grid size, obeys conservation laws.
CONSERVATIVE LOADA load that always acts in a fixed direction regardless of the deformation of the body, for example, gravity.
CONSERVATIVENESSThe property of a numerical scheme in which the laws of conservation are adhered to.
CONSISTENCYThe property of a numerical scheme in which the algebraic equations produced by the discretisation process are equivalent to the original governing equations as the grid spacing tends to zero.
CONSISTENT DISPLACEMENTS AND FORCESThe displacements and forces act at the same point and in the same direction so that the sum of their products give a work quantity. If consistent displacements and forces are used the the resulting stiffness and mass matrices are symmetric.
CONSISTENT TANGENT STIFFNESS METHODA technique in plasticity analysis using stiffnesses at each iteration that accurately incorporates the current state of plasticity.
CONSTANT STRAIN CONSTANT STRESSFor structural analysis an element must be able to reproduce a state of constant stress and strain under a suitable loading to ensure that it will converge to the correct solution. This is tested for using the patch test.
CONSTITUTIVE EQUATIONA description of any linear or non-linear material behaviour law, usually relating strain, stress and temperature.
CONSTITUTIVE RELATIONSHIPSThe equations defining the material behaviour for an infinitesimal volume of material. For structures these are the stress-strain laws and include Hookes law for elasticity and the Prandle-Reuss equations for plasticity.
CONSTRAINED METHODSNon-linear solution procedures in which the solution is constrained to follow a certain path during the iteration process, e.g. arc length methods (q.v.).
CONSTRAINT EQUATIONS (MULTI POINT CONSTRAINTS)If one group of variables can be defined in terms of another group then the relationship between the two are constraint equations. Typically the displacements on the face of an element can be constrained to remain plane but the plane itself can move.
CONSTRAINTSFixed relationships between the basic degrees of freedom in a finite element model.
CONTACT DISCONTINUITYA discontinuity across which density and tangential velocity may be discontinuous but pressure and normal velocity are constant and there is no mass transfer. The best known discontinuity found in CFD is due to shock waves.
CONTACT ELEMENTS/GAP ELEMENTSElements, as lines or areas, used to model states of contact between surfaces.
CONTACT INSTABILITYThis occurs in contact analysis when instabilities are generated due to local mesh density and hourglassing. They can cause convergence problems.
CONTACT PROBLEMSA contact problem occurs when two bodies that are originally apart can come together, or two bodies that are originally connected can separate.
CONTINUITYA system that exhibits continuity and expresses conservation (generally of mass).
CONTINUOUS MASS MODELSThe system mass is distributed between the degrees of freedom in a kinematically equivalent manner. The mass matrix is not diagonal.
CONTINUOUS MODELSThe model is defined in terms of partial differential equations rather than in finite degree of freedom matrix form.
CONTINUUM REGION ELEMENT (CRE) METHODA single element test where the element is defined within a region where there is a known stress field. Point loads and nodal displacements can then be calculated and applied over the element, whose shape can vary at will, to test the element’s response.
CONTOUR PLOTSA representation of a surface showing lines of constant value for a particular variable such as temperature (isotherms) or pressure (isobars). The regions between the lines are often filled to produce continuously coloured plots representing variable values.
CONTOUR PLOTTINGA graphical representation of the variation of a field variable over a surface. A contour line is a line of constant value for the variable. A contour band is an area of a single colour for values of the variable within two limit values.
CONTRAVARIANT COMPONENTSVector components projected normal to co-ordinate surfaces. See covariant components.
CONTROL POINTSPoints at which discretised equations are solved and variable values are obtained.
CONTROL SURFACEThe bounding surface of a control volume.
CONTROL VOLUMEThe volume over which the partial differential equations describing fluid flow are integrated to obtain discretised (algebraic) equations.
CONTROL VOLUME METHODA numerical solution method in which the domain is divided into a finite number of control volumes. The governing equations are then discretised and solved for the individual volumes as part of the whole.
CONVECTED COORDINATE FORMULATION (ALSO CALLED CO-ROTATIONAL FORMULATION)A geometrically non-linear formulation in which a local cartesian coordinate system is attached to the element and is allowed to continuously translate and rotate with the element during deformation.
CONVECTIONA mode of heat transfer between a fluid and solid boundary. The heat energy is transferred by the movement of macroscopic fluid particles.
CONVERGENCEProperty of a numerical method to tend towards a single answer. For any non-linear solution procedure, convergence is achieved when sufficient iterations within a given increment of time or load have produced an equilibrium state to within a given convergence criterion.
CONVERGENCE CRITERIONCriterion by which a solution is judged to determine if it is sufficiently converged. Convergence is normally dependent on satisfaction of a number of such criteria.
CONVERGENCE ERRORThe difference between the iterative and exact solutions of the discretised equations.
CONVERGENCE REQUIREMENTSFor a structural finite element to converge as the mesh is refined it must be able to represent a state of constant stress and strain free rigid body movements exactly. There are equivalent requirements for other problem types.
CONVOLUTION INTEGRAL (DUHAMEL INTEGRAL)The integral relating the dynamic displacement response of the structure at any time t to the forces applied before this time.
CO-ORDINATE STRETCHINGA grid generation method involving stretching of the grid in one co-ordinate direction or more.
COORDINATE SYSTEMThe set of displacements used to define the degrees of freedom of the system.
CORRECTED VISCOSITY SCHEMEA scheme used to improve the accuracy of the Lax-Friedrichs scheme.
CORRECTION FORMULAEApproximation of flow variables by the sum of a guessed value and a correction value.
CORRECTOR STEPAdditional step used to improve on a guessed set of values (used in pressure- velocity coupling methods such as SIMPLE, PISO, etc.).
CORRESPONDING FORCES AND DISPLACEMENTSA force and a displacement are said to correspond if they act at the same point and in the same direction. Forces and translational displacements can correspond as can moments and rotations. Corresponding forces and displacements can be multiplied together to give a work quantity. Using corresponding forces and displacements will always lead to a symmetric stiffness matrix.
COUETTE FLOWA flow driven solely by boundary movement in which there is no pressure gradient. Hence an analytical solution is possible (see NAFEMS CFD Workbook of Examples).
COULOMB DAMPING (ALSO CALLED DRY FRICTION DAMPING)A damping model in which the damping force is a constant and always opposes the velocity of motion.
COUPLED PARTICLE FLOWFlow of discrete particles, bubbles or drops in a continuum in which the movement of the particles influences the flow of the continuum and vice versa.
COUPLED PROBLEMSThese occur when multiple geometric domains are to be linked or when different physical states are to be solved, in each case in a dependent manner.
COUPLED SOLVERA solver that typically solves for continuity, momentum and energy (and potentially species) simultaneously. It is an alternative to a segregated solver and is often used for compressible flows.
COURANT NUMBERThe speed of sound, multiplied by the ratio of the time step length to the cell length. This ratio is the time required for a quantity or fluid particle to be convected through a small distance. Therefore, the Courant number can be viewed as a time step to convection time scale ratio.
COURANT-FRIEDRICHS-LEWY (CFL) CONDITIONSee CFL condition
COVARIANT COMPONENTSFor simulating fluid flow and heat transfer in complex geometries, the governing equations are expressed in generalised curvilinear co-ordinates in which the dependent variable can be cartesian, covariant or contravariant velocity components. The covariant velocity components align with the curvilinear co- ordinates but are not orthogonal to the cell faces. It has the advantage that the cross pressure gradient terms in the momentum equation disappear.
CRACK CLOSURE WORK METHODSThese calculate the energy release rate by two finite element calculations, calculating the point force needed to either open or close the crack over a short length after the first run, and equating this work done to the required energy change; several variants exist.
CRACK ELEMENT (CRACK TIP ELEMENT)An element that includes special functions to model the stress field at the tip of a crack. This is commonly achieved by using quadratic elements with mid side nodes at the quarter chord points.
CRACK PROFILE OR FRONTThe sharp end of a crack inside a three dimensional body, which is a curve of known position and of finite length, and which can vary with time. Any two dimensional section cutting this crack profile will contain a part of the crack ending in a crack tip (q.v.), and is frequently studied to give simplified values of the fracture parameters of interest.
CRACK PROPAGATIONThe relatively steady growth of cracks, usually during the fatigue life of a product. It could also be due to non-linear material degradation such as ductile void growth and coalescence.
CRACK PROPAGATION (FRACTURE MECHANICS)The process by which a crack can propagate through a structure. It is commonly assumed that a crack initiates when a critical value of stress or strain is reached and it propagates if it can release more than a critical amount of energy by the crack opening.
CRACK TIPThe sharp end of a crack inside a given two dimensional body, at a point whose position is known and which may move over time.
CRACK TIP ELEMENTSFinite elements sited around crack tips, modified to contain displacement variations representing the singular strain fields that exist there, thereby giving greater accuracy than the standard polynomial variations.
CRACK TIP EQUATIONSThese are mathematical equations which are valid for elastic crack tip conditions, relating components of stress and displacement with local geometric position relative to the crack tip. The equations give the stress intensity factors.
CRACK TIP OPENING DISPLACEMENT (CTOD)This is a measure of how much the crack tip opens up under load when significant plastic deformation occurs in that region. It is useful as a fracture parameter.
CRANK-NICHOLSON SCHEMEA method for numerically integrating first order dynamic equations of motion. It is widely used as a technique for solving thermal transient problems.
CRANK-NICOLSON SCHEMEA semi-implicit solution scheme for unsteady flows.
CREEP LAWSThe laws that govern time dependent creep, based on simple experimental tests. Typical laws are those of Norton, Prandtl, and Bailey.
CREEP STRAINIrrecoverable permanent strain due to time dependent creep.
CRITICAL CONDITIONCondition at which the nature of a flow changes, e.g. from laminar to turbulent or where a shock wave is produced.
CRITICAL DAMPINGThe damping value for which the impulse response is just oscillatory.
CRITICAL ENERGY RELEASEThis is a material property defining the minimum energy that a propagating crack must release in order for it to propagate. Three critical energies, or modes of crack propagation, have been identified. Mode 1 is the two surfaces of the crack moving apart. Mode 2 is where the two surfaces slide from front to back. Mode 3 is where the to surfaces slide sideways.
CRITICAL VALUESThese are numerical quantities representing the various fracture parameters, at those levels of load that cause some relevant fracture event to happen. For example, the critical value of the stress intensity factor is the fracture toughness.
CRITICALLY DAMPED SYSTEMThe dividing line between under damped and over damped systems where the equation of motion has a damping value that is equal to the critical damping.
CRITICALLY DAMPED SYSTEM CRITICAL DAMPINGThe dividing line between under damped and over damped systems where the equation of motion has a damping value that is equal to the critical damping.
CURVATURE COMPENSATED CONVECTIVE TRANSPORTSee CCCT.
CURVILINEAR GRIDA grid based on curvilinear co-ordinates.
CYCLIC BOUNDARY CONDITIONA boundary condition in which conditions at one surface of the calculation domain are assumed continuous with those at another, employed for cyclically repeating flows. Also sometimes known as a periodic boundary condition.
CYCLIC GRIDA cyclic grid repeats in a cyclic manner.
CYCLIC LOADINGLoads that repeatedly oscillate between maximum and minimum values over time.
CYCLIC SYMMETRYA generalisation of axisymmetry. The structure is composed of a series of identical sectors that are arranged circumferentially to form a ring. A turbine disc with blades attached is a typical example.
CYLINDRICAL CO-ORDINATESCo-ordinates based on a length, radius and angle.