# The NAFEMS Glossary

# Terms S-Z

Safe LifeA design philosophy in which products are designed to survive a specific operational life with a chosen reserve.

Sandwich StructureA composite structure composed of lightweight core material (usually honeycomb or foam) to which two relatively thin, dense, high-strength, functional or decorative skins are adhered.

ScalarsQuantities that have no direction associated with them, e.g. temperatures. Scalar problems only have one degree of freedom at a node. See also Vectors.

Secant StiffnessThe stiffness defined by the slope of the line from the origin to the current point of interest on a load/deflection curve.

Second Piola-Kirchhoff StressThe work conjugate stress measure to the Green’s Strain.

Secondary ComponentsComponents of a structure not of direct interest but they may have some influence of the behaviour of the part of the structure that is of interest (the primary component) and have to be included in the analysis in some approximate form.

Secondary CreepThat part of a creep test where the strain rate is constant.

Seepage FlowFlows in porous materials.

Seismic AnalysisThe calculation of the dynamic displacement and stress response arising from earthquake excitations.

Selected Reduced IntegrationA form of Gaussian quadrature where different sets of Gauss Points are used for different strain components.

Self Adjoint EquationsA form of matrix products that preserves symmetry of equations. The product A x B x A(transpose) is self-adjoint if the matrix B is symmetric. The result of the product will be symmetric for any form of A that is of a size compatible with B. This form of equation occurs regularly within the finite element method. Typically it means that for a structural analysis the stiffness (and mass) matrices for any element or element assembly will be symmetric.

Self Equilibrating LoadsA load set is self equilibrating if all of its resultants are zero. Both translation and moment resultants are zero.

Semi-Loof ElementA form of thick shell element.

ShakedownOccurs in cyclic loading where the plastic strain in each cycle stabilises so that the total strain within a cycle is less than twice the yield strain (the strain when the stress reaches the yield stress).

Shape FunctionsEquations which are used to define the variation of the geometry and main degrees of freedom (typically displacement) within an element: the equations vary over different element types.

Shape ParametersWays of defining an element’s shape, with particular reference to how the shape differs from the theoretically perfect shape for that element type. Parameters include Aspect Ratios, taper, skew, curvature, warpage and variation of the Jacobian.

Shape SensitivitySee Distortion.

Shear LockingThe phenomena which occurs when thick elements give overstiff results when modelling thin beams/plates/shells, due to an excess of shear energy being present. It can also affect 2D and 3D continuum elements.

Simpson's RuleA method for numerically integrating a function.

Simultaneous Vector IterationA method for finding the first few eigenvalues and eigenvectors of a finite element system. This is also known as Subspace Vector Iteration.

Single Degree Of FreedomThe system is defined by a single force/displacement equation.

Single Element TestsAny test of an element’s performance using only one element. See also Patch Test and Continuum Region Element (CRE) method.

Single Point ConstraintWhere the constraint is unique to a single node point.

Singular MatrixA square matrix that cannot be inverted.

Skew Distortion (Angular Distortion)A measure of the angular distortion arising between two vectors that are at right angles in the basis space when these are mapped to the real coordinate space. If this angle approaches zero the element becomes ill- conditioned.

Skew Symmetric MatrixA matrix is Skew Symmetric if the ij term is equal to minus the ji term. All of the diagonal terms are zero. See also Hermitian Matrix.

Skyline MatrixSee Variable Bandwidth Matrix.

SlidingIn contact analysis, when adjacent surfaces move tangentially to one another.

Smeared Crack ModelIn the non-linear analysis of concrete structures, a model which does not follow discrete cracks, but assumes damage is caused by closely spaced cracks associated with an integration point.

Snap Back BucklingA situation that occurs when a vertical line in the load-displacement curve is encountered, and two or more equilibrium states are possible for the same applied load. Also referred to as Snap Through Buckling.

Snap Through BucklingA situation that occurs when a vertical line in the load-displacement curve is encountered, and two or more equilibrium states are possible for the same applied load. Also referred to as Snap Back Buckling.

SofteningIn plastic flow, this is a contraction of the yield surface that leads to localisation phenomena.

Solid ElementsThree dimensional continuum elements.

Solution AccuracyThe accuracy of the solution of the equations used in the finite element method, usually referring to the main stiffness equations. When a very large number of variables exist or the model generated is poor, accuracy can be lost due to ill-conditioning arising from the numerical processes.

Solution DiagnosticsMessages that are generated as the finite element solution progresses. These should always be checked for relevance but they are often only provided for information purposes

Solution EfficiencyAn indication of the efficiency of the solution of the equations used in the finite element method, usually referring to the main stiffness equations. Minimising the number of such equations without compromising solution accuracy is a common challenge.

Sparse Matrix MethodsSolution methods that exploit the sparse nature of finite element equations. Such methods include the Frontal Solution and Cholesky (skyline) factorisation for direct solutions, conjugate gradient methods for iterative solutions and the Lanczos method and subspace iteration (simultaneous vector iteration) for eigenvalue solutions.

Spectral DensityThe Fourier transform of the correlation function. In random vibrations it gives a measure of the significant frequency content in a system. White Noise has a constant spectral density for all frequencies.

Spline CurvesA curve fitting technique that preserves zero, first and second derivative continuity across segment boundaries.

Spurious CracksCracks that appear in a mesh when the elements are not correctly connected together. This is usually an error in the mesh generation process.

Static CondensationSee Condensation.

Statically Determinate StructureA structure where all of the unknowns can be found from equilibrium considerations alone.

Statically Equivalent LoadsEquivalent nodal loads that have the same equilibrium resultants as the applied loads but do not necessarily do the same work as the applied loads.

Statically IndeterminateA structure where all of the unknowns can not be found from equilibrium considerations alone. The compatibility equations must also be used. In this case the structure is said to be redundant.

Stationary Random ExcitationA force or response that is random but its statistical characteristics do not vary with time.

Steady State Creep LawA creep model in which there are no hardening or softening effects.

Steady State ResponseThe response of the system to a periodic forcing function when all of the transient components of the response have become insignificant.

STEPA series of computer readable data models which form an international standard for exchange of product definition data, relevant to finite elements as a medium for data transfer to and from CAD packages. It is planned to eventually replace the existing standards such as IGES, SET and VDA-FS.

Step-By-Step IntegrationMethods of numerically integrating time varying equations of motion. These methods can take the form of an Explicit Solution Scheme or an Implicit Soltuion Scheme.

Stiffness MatrixThe parameter(s) that relate the displacement(s) to the force(s). For a discrete parameter multi degree of freedom model this is usually given as a stiffness matrix.

Strain EnergyThe energy stored in the system by the stiffness when it is displaced from its equilibrium position.

Strain Energy Release RateFor a hypothetically small increase in crack length or area, this is the amount of strain energy released divided by that length or area. It equals the negative of the Potential Energy Release Rate when elastic conditions predominate.

Strain Hardening LawUsed in analysing creep behaviour under variable load where the creep strain rate is assumed to depend on the current stress and accumulated creep strain, or in plasticity where the current yield stress is a function of the plastic strain.

Strain-Life ApproachIn fatigue, a method whereby the predicted life of a product is based on calculated strain values, typically used in low cycle fatigue. See also Stress-Life Approach.

Stress AveragingThe process of filtering the raw finite element stress results to obtain the most realistic estimates of the true state of stress. Also referred to as Stress Smoothing.

Stress ConcentrationA local area of the structure where the stresses are significantly higher than the general stress level. A fine mesh of elements is required in such regions if accurate estimates of the stress concentration values are required.

Stress DiscontinuitiesLines along which the stresses are discontinous. If the geometry or loading changes abruptly along a line then the true stress can be discontinous. In a finite element solution the element assumptions means that the stresses will generally be discontinuous across element boundaries. See also Stress Error Estimates.

Stress Error EstimatesIn a finite element solution the element assumptions means that the stresses will generally be discontinuous across element boundaries. The degree of discontinuity can then be used to form an estimate of the error in the stress within the finite element calculation. See also Stress Discontinuities.

Stress ExtrapolationThe process of taking the stress results at the optimum sampling points for an element and extrapolating these to the element node points.

Stress Intensity FactorA fracture parameter at a crack tip when under conditions of Linear Elastic Fracture Mechanics (LEFM). It is a function of applied load and crack length, suitably dimensioned to have a finite value at the tip even though the stresses are singular there, and may be used to characterise the state of fracture there.

Stress RelaxationOccurs in creep problems when the structure is loaded up to a certain stress level and then held at constant strain.

Stress SmoothingThe process of filtering the raw finite element stress results to obtain the most realistic estimates of the true state of stress. Also referred to as Stress Averaging.

Stress StiffnessSee Geometric Stiffness.

Stress Substitution MethodA method of calculating the Stress Intensity Factor at a given crack tip using the local stresses from FE analysis and known crack tip equations.

Stress WavesElastic stresses that propagate through materials at high speeds due to impact loads.

Stress / Strain VectorThe stress / strain vector is the components of stress / strain written as a colunn vector. For a general three dimensional body this is a (6x1) matrix. See also Stress / Strain Tensor.

Stress / Strain TensorThe components of stress (strain) written in tensor form. For a general three dimensional body this forms a (3x3) matrix with the direct terms down the diagonal and the shear terms as the off-diagonals. See also Stress / Strain Vector.

Stress-Life ApproachIn fatigue, a method whereby the predicted life of a product is based on calculated stress values, typically at stress concentrations and for high cycle fatigue. See also Strain-Life Approach.

Stress-Strain LawThe material property behaviour relating stress to strain. For a linear behaviour this is Hookes law (linear elasticity). For elastic plastic behaviour it is a combination of Hookes law and the Prandtl-Reuss equations.

Structured Grid (Or Mesh)A grid (in CFD) or mesh where the elements form a regular pattern.

Subspace Vector IterationA method for finding the first few eigenvalues and eigenvectors of a finite element system. This is also known as Simultaneous Vector Iteration.

Substructure (Also Called Superelement)A mesh modelling technique whereby a part of a structure, containing a number of elements, can be stored by the software as a single element. It can then be used for a variety of different purposes, just as if it were a new element type with its own stiffness matrix.

Substructuring MethodSubstructuring is a form of equation solution method where the structure is split into a series of smaller structures - the substructures. These are solved to eliminate the internal freedoms and the complete problem solved by only assembling the freedoms on the common boundaries between the substructures. The intermediate solution where the internal freedoms of a substructure have been eliminated giving the super element matrix for the substructure. Also referred to as the Super Element Method.

Super Element MethodSubstructuring is a form of equation solution method where the structure is split into a series of smaller structures - the substructures. These are solved to eliminate the internal freedoms and the complete problem solved by only assembling the freedoms on the common boundaries between the substructures. The intermediate solution where the internal freedoms of a substructure have been eliminated giving the super element matrix for the substructure. Also referred to as the Substructuring Method.

SuperpositionFor a linear system the response is the same if it is found by adding together two or more separate forcing functions and then solving the equations or by solving for the separate forcing functions and then adding the responses together. The second method of solving for each forcing function and adding the response is superposition.

SupportsDegrees of freedom where the variable is known before the solution is found. Typically the zero displacements at fixed points in a structural analysis or the points of known temperature in a heat conduction analysis. Generally there must be some points of known value (i.e the structure must be supported) before the equations can be solved.

Surface ElementSpecial elements that are used to model surface boundary conditions. Typically surface heat transfer elements used to model surface heat transfer coefficients in heat conduction problems.

Symmetrical MatrixA matrix is symmetric if it is square and if the ij term is equal to the ji term. See also Skew Symmetric Matrix if it is square and if the ij term is equal to minus the ji term. All of the diagonal terms are zero. A matrix is Hermitian if it is square, the real part is symmetric and the imaginary part is skew symmetric.

Symmetry (Of Model)In constructing a finite element model, the meshing of similar shapes with similar loading within the model can be avoided by using the principles of symmetry, and by using suitable boundary conditions. The different types of symmetry include: repetitive, mirror, axial and cyclic. Asymmetric loading can also be modelled with suitable boundary conditions.

Tangent StiffnessFor non-linear problems this is the slope of the load/deflection curve for the current solution position.

Tangent Stiffness MatrixThe matrix of coefficients corresponding to the derivatives of the residual forces with respect to the displacement degrees of freedom: this matrix is evaluated and factorised during the incremental-iterative solution procedure.

Tertiary CreepThat part of a creep test where the strain rate is increasing.

Tetrahedron (Tetrahedral Element)A three dimensional four sided solid element.

Thermal CapacityThe material property defining the thermal inertia of a material. It relates the rate of change of temperature with time to heat flux.

Thermal ConductivityThe material property relating temperature gradient to heat flux.

Thermal ContactThe analysis of contacting surfaces when thermal effects are significant.

Thermal LoadsThe equivalent loads on a structure arising from thermal strains. These in turn arise from a temperature change.

Thermal StrainsThe components of strain arising from a change in temperature.

Thick Shell ElementIn a shell element the geometry is very much thinner in one direction than the other two. It can then be assumed stresses can only vary linearly at most in the thickness direction. If the transverse shear strains are not ignored then a thick shell model is formed. This uses the Mindlin shell theory. For the finite element method the thick shell theory generates the most reliable form of shell elements. There are two forms of such elements, the Mindlin shell and the Semi-Loof shell. See also Thin Shell Element.

Thin Shell ElementIn a shell element the geometry is very much thinner in one direction than the other two. It can then be assumed stresses can only vary linearly at most in the thickness direction. If the through thickness shear strains can be taken as zero then a thin shell model is formed. This uses the Kirchoff shell theory. See also Thick Shell Element.

Time DomainThe structures forcing function and the consequent response is defined in terms of time histories. The Fourier transform of the time domain gives the corresponding quantity in the frequency domain.

Time Hardening LawUsed in analysing creep behaviour under variable load where the creep strain rate is assumed to depend on the current stress and time from the start of the test.

Time Stepping SchemesMethods for integrating the governing equations of time dependent non-linear problems. Examples include Newmark’s family of methods for solving the transient dynamic equilibrium equations and time marching procedures for creep analysis. See also Newmark Method.

Total Lagrangian FormulationIn geometrically non-linear analysis, a formulation in which all static and kinematic variables are referred to the initial undeformed configuration. See also Updated Lagrangian Formulation.

Trace Of The MatrixThe sum of the leading diagonal terms of the matrix.

Transfinite MappingA systematic method for generating element shape functions for irregular node distributions on an element.

Transformation MethodSolution techniques that transform coordinate and force systems to generate a simpler form of solution. The Eigenvectors can be used to transform coupled dynamic equations to a series of single degree of freedom equations.

Transient AnalysisAn analysis is transient when at least one of the parameters involved in the boundary conditions, material properties or loading conditions is time dependent.

Transient ForceA forcing function that varies for a short period of time and then settles to a constant value.

Transient ResponseThe response of a system to applied forces that are of short duration compared to the periods of the Resonant Frequency of the system.

Transition ElementSpecial elements that have sides with different numbers of nodes. They are used to couple elements with different orders of interpolation, typically a transition element with two nodes on one edge and three on another is used to couple a 4-node quad to an 8-node quad.

Tresca Equivalent StressSee Von Mises Equivalent Stress.

Tresca Yield CriterionIs used for metals and assumes that yielding starts when the maximum value of the shear stress reaches a given value. It gives a hexagonal cylindrical shape in principal stress space.

Triangular ElementsTwo dimensional or surface elements that have three edges.

True StrainA particular strain measure used in large strain elasto-plasticity; the log of one plus the Engineering Strain, or the integral of the incremental change of length over the current length.

True StressThe force divided by the current (instantaneous) area. See also Cauchy Stress.

TurbulenceA chaotic state of fluid motion where the velocity and pressure at a point change continuously with time.

Ultimate StressThe failure stress (or Equivalent Stress) for the material.

Uncertainty QuantificationFormulation of a statistical model to characterise imperfect and/or unknown information in engineering simulation and physical testing for predictions and decision making.

Undamped Natural FrequencyThe square root of the ratio of the stiffness to the mass (the square root of the Eigenvalues). It is the frequency at which an undamped system vibrates naturally. A system with n degrees of freedom has n natural frequencies.

Under-damped SystemA system that has an equation of motion where the damping is less than critical. It has an oscillatory impulse response. See also Critical Damping.

Unit MatrixA diagonal matrix with unit values down the diagonal.

Unstructured Grid (Or Mesh)A grid (in CFD) or mesh where the elements form no regular pattern.

Updated Lagrangian FormulationIn geometrically non-linear analysis, a formulation in which all static and kinematic variables are referred to the last calculated configuration. See also Total Lagrangian Formulation.

Upper Bound SolutionThe assumed stress form of the finite element solution gives an upper bound on the maximum stresses and strain energy (i.e. these are over estimated) for a given set of displacements. See also Strain Energy.

Upwinding In FluidsA special form of weighting function used in viscous flow problems (solution to the Navier-Stokes equations) used in the weighted residual method to bias the results in the direction of the flow.

ValidationThe process of determining the degree to which a model is an accurate representation of the real world from the perspective of the intended uses of the model.

Variable Bandwidth MatrixA sparse matrix where the bandwidth is not constant. Some times called a Skyline Matrix.

VelocityThe first time derivative of the displacement.

VerificationThe process of determining that a computational model accurately represents the underlying mathematical model and its solution.

Virtual Crack Extension MethodA method for calculating fracture criteria at a crack tip using the potential energy change with crack growth and utilising in an efficient manner certain characteristics of the stiffness equations.

Virtual DisplacementsAn arbitrary imaginary change of the system configuration consistent with its constraints.. See also Virtual Work.

Virtual ForcesSee Virtual Work.

Virtual WorkTechniques for using work arguments to establish equilibrium equations from compatibility equations (Virtual Displacements) and to establish compatibility equations from equilibrium (virtual forces).

Visco-ElasticityA non-linear material behaviour in which both the effects of elasticity and creep are exhibited, so that the stress is dependent on the strain rate.

Visco-PlasticityA non-linear material behaviour theory in which time rate effects are included in the plastic deformation process; thus, stresses and strains describing the plastic state are also time dependent.

Viscous DampingThe damping is viscous when the damping force is proportional to the velocity.

Viscous Damping MatrixThe matrix relating a set of velocities to their corresponding velocities.

Volume DistortionSee Volumetric Distortion.

Volumetric DistortionThe distortion measured by the determinant of the Jacobian matrix, det j.

Volumetric StrainSee Hydrostatic Stress And Strain.

Volumetric StressSee Hydrostatic Stress.

Von Mises Equivalent StressEquivalent stress measures to represent the maximum shear stress in a material. These are used to characterise flow failures (e.g. plasticity and creep). From test results the Von-Mises form seems more accurate but the Tresca form is easier to handle.

Von Mises StressThe second invariant of the deviatoric stress tensor. This is a scalar value and is used to relate a 3D stress field to a 1D tensile test. Thus, it is often called an effective or Equivalent Stress.

Von Mises Yield CriterionIs used to describe the yield of metals and assumes that yielding commences when the von Mises stress reaches a critical value.

VectorsVector quantities have a direction associated with them, e.g. displacements. Vector problems have more than one degree of freedom at a node.See also Scalars.

Wave PropagationThe dynamic calculation involving the prediction of the history of stress and pressure waves in solids and fluids.

Wavefront (Front)The wavefront of a symmetric matrix is the maximum number of active nodes at any time during a Frontal Solution process. It is a measure of the time required to factorise the equations in a frontal solution. It is minimised be element renumbering.

Wavelength SolutionSee Frontal Solution.

Weighted ResidualsA technique for transforming a set of partial differential equations to a set of simultaneous equations so that the solution to the simultaneous equations satisfy the partial differential equations in a mean sense.The form used in the finite element method is the Galerkin Method. This leads to identical equations to those from virtual work arguments.

Whirling StabilityThe stability of rotating systems where centrifugal and Coriolis are also present.

White NoiseWhite noise has a constant Spectral Density for all frequencies.

Wilson Theta MethodAn implicit solution method for integrating second order equations of motion. It can be made unconditionally stable.

Word LengthWithin a digital computer a number is only held to a finite number of significant figures. A 32bit (single precision) word has about 7 significant figures. A 64bit (double precision) word has about 13 significant figures. All finite element calculations should be conducted in double precision.

Yield CriterionIn the theory of plasticity, a law defining the limit of elastic behaviour under any possible combination of the stress components at any point: the criteria of Tresca and Von Mises Equivalent Stress are common for metals.

Yield StressThe stress level at which yielding commences in a uniaxial stress-strain state.

YieldingThe transition of material behaviour from elastic to plastic.

Youngs ModulusThe material property relating a uniaxial stress to the corresponding strain.

Zero Energy ModesSpurious element deformations that occur with zero strain energy, due to particular numerical integration schemes. See alsoHourglass Effects