# The NAFEMS Glossary

# 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.

Damped EigenvaluesSee also Complex Eigenvalues.

Damped EigenvectorsSee also 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 FactorThe 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.

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.

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 Stress And StrainRepresents the shear component of stress and strain, i.e. the remainder after deducting the Hydrostatic Stress or Strain. The deviatoric components govern plastic and creep flows, where there is change in shape but not of volume. See also Hydrostatic Stress And Strain.

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 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

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.

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.

Direction CosinesThe cosines of the angles a vector makes with the global x,y,z axes.

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.

Displacement ControlWhen displacements are selected as the controlling parameter in a non-linear solution (as opposed to load or time control). See also Non-Linear Analysis.

Displacement Method (Displacement Solution)A form of discrete parameter model where the displacements of the system are the basic unknowns.Non-Linear Analysis.

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.

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. See also Shape Sensitivity.

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 Method.

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.

Dynamic AnalysisAn analysis that includes the effect of the variables changing with time as well as space.

Dynamic ContactThe analysis of contacting surfaces when inertia effects cannot be ignored.

Dynamic FlexibilityThe factor relating the steady state displacement response of a system to a sinusoidal force input. Also see Receptance.

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 Stiffness MatrixIf the structure is vibrating steadily at a frequency w then the dynamic stiffness matrix is (K+iωC-ω2M).

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.

Effective StrainA scalar quantity defined (usually as the von Mises strain) to represent the individual strain components at any reference point also used for strain rates. See also Equivalent Strain.

Effective StressA scalar quantity defined (usually as the von Mises stress, q.v.) to represent the individual stress components at any reference point. See also Equivalent Stress.

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 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. See also Latent Roots.

EigenvectorsThe 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. See also Latent Vectors.

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 Mechanics (EPFM)See Post Yield Fracture Mechanics (PYFM).

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 GenerationThe process of generating a mesh of elements over the structure. This is normally done automatically or semi-automatically. This is also referred to as Mesh Generation.

Element Shape ParameterA mathematical parameter to quantify the quality of an element for a particular purpose.

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.

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 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 NAFEMS Engineering Simulation Quality Standard that is being developed by the Simulation Governance and Management Working Group defines Engineering Simulation as the whole process of physics modelling of the engineering system, calculation by analytical or numerical methods, verification, validation and interpretation of results in terms of the physical behaviour of the engineered system. An alternative definition is that engineering simulation is the use of numerical, physical or logical models of systems and scientific problems in predicting their response to different physical conditions. The

Engineering StrainThe ratio of the change in length over a given length to the original length. Also see Nominal Strain.

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.

Eulerian FormulationA geometrically non-linear formulation where the equilibrium conditions are evaluated in the deformed configuration. Also see Lagrangian Formulation.

Eulerian 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. See also Lagrangian 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.

Explicit MethodsThese are a number of 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. See also Implicit Methods.

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. It is used in both static and dynamics analyses. See also Implicit Methods.

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.

Fast Fourier TransformA method for calculating Fourier transforms that is computationally very efficient.

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 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 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. See also Galerkin Method.

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.

Flow RuleUsed in plasticity to define a relationship between the plastic strain increment and the stress increment. See also Normality Rule.

Fluidity ParameterIn elastic-viscoplastic analysis, a parameter used in the evaluation of the viscoplastic strain rate. See also Visco-Plasticity.

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 the system.

Fourier ExpansionsFunctions that repeat themselves in a regular manner can be expanded in terms of a Fourier series.

Fourier SeriesSee Fourier Expansions.

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.

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. , Crack Tip Opening Displacement

Fracture Toughness (KIC )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 SolutionA form of solving the finite element equations using Gauss elimination that is very efficient for the finite element form of equations.

Froude NumberA fluid flow measure of the ratio of inertia forces to gravitational forces, typically used in free surface flows.

Galerkin MethodGalerkin Methods are on of the class of Weighted Residual Methods used for converting a continuous operator problem into a discrete problem. See also Weighted Residuals.

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/Gap Elements.

Gauss Point ExtrapolationSee Gauss Point Stresses.

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. Also see Barlow Points.

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.

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 IntegrationA 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 QuadratureSee Gaussian Integration.

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. See also Eigenvectors.

Generalised MassThe mass associated with a generalised displacement.

Generalised StiffnessThe stiffness associated with a generalised displacement.

Geometric 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.

Global Stiffness MatrixThe assembled stiffness matrix of the complete structure.

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.

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 (subspace 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. See also Subspace Vector Iteration.

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.

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.

Hermitian MatrixA matrix is Hermitian if it is square, the real part is symmetric and the imaginary part is skew symmetric. See also Symmetrical Matrix.

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.

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. See also Zero Energy Modes.

Hourglass ModesZero energy modes of low order quadrilateral and brick elements that arise from using reduced integration. These modes can propagate through the complete body.

H-RefinementMaking the mesh finer over parts or all of the body is termed h-refinement. Making the element order higher is termed p-refinement. See also Mesh Density.

Hybrid CompositeA composite with two or more types of reinforcing fibres.

Hybrid ElementsElements that use stress interpolation within their volume and displacement interpolation around their boundary.

Hydrostatic 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. See also Deviatoric Stress and Strain.

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.

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.

Ill-Conditioning ErrorsNumerical (rounding) errors that arise when using ill-conditioned equations.

Ill-ConditioningEquations 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-Conditioned EquationsSee Ill-Conditioning.

Implicit MethodsThese are a number of methods for integrating equations of motion. Implicit methods can deal with mildly non-linear problems but with large steps. See also Explicit Methods.

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. It is used in both static and dynamic analyses. See also Explicit Solution Scheme.

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 and so remains constant, with a Mach number below approximately 0.3. See also Mach Number.

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.

Inelastic Material BehaviourA material behaviour where residual stresses or strains can remain in the body after a loading cycle, typically plasticity and creep.

InertanceThe ratio of the steady state acceleration response to the value of the forcing function. See also Accelerance.

Inertia ForceThe force that is equal to the mass times the acceleration.

Initial BucklingThe load at which a structure first buckles.

Initial Stiffness MethodA modified Newton-Raphson solution in which the initial linearly elastic, small displacement, stiffness matrix is used throughout the analysis. See also Newton-Raphson Method.

Initial StrainsThe components of the strains that are non-elastic. Typically thermal strain and plastic strain.

Integration By PartsA method of integrating a function where high order derivative terms are partially integrated to reduce their order.

Interpolation 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.

Isoparametric ElementsElements in which the displacements and geometry variation within an element are represented by the same Shape Functions.

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. See also Kinematic Hardening.

Isotropic MaterialMaterials where the material properties are independent of the co-ordinate system.