# The NAFEMS Glossary

# Terms A-C

AcceleranceThe ratio of the steady state acceleration response to the value of the forcing function. See also Inertance.

AccelerationThe second time derivative of the displacement (the first time derivative of the velocity).

AdaptivitySee Mesh Adaptivity.

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.

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 when there is a progressive increase in total strain with each cycle. See also Cyclic Loading.

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.

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. See also displacement control.Displacement Control.

Area CoordinatesA special coordinate system that is used for defining shape functions for triangular and tetrahedral elements.

Aspect RatiosThe ratio of the different element side or edge lengths, used for establishing amounts of distortion. Generally bodies with high aspect ratios (long and thin) are more ill-conditioned for numerical solution than bodies with an aspect ratio of one. See alsoDistortion.

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.

Assumed Displacement MethodThe assumed displacement method determines the vector of internal displacement by assuming that they relate to the nodal displacement vector via a set of Shape Functions.

Augmented Lagrangian MethodA combination of the penalty function and Lagrange multiplier methods. Used in contact analysis, where the contact force is defined in terms of the Lagrange multiplier plus a penalty stiffness term. See also Lagrange Multipliers.

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.

Axisymmetric ElementAn element defined by rotating a cross-section about a centre line.

Axisymmetric Thin ShellAn element forms an axisymmetric thin shell if a line element is rotated about an axis.

Axisymmetrical Thick ShellAn 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.

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 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 ElementA line element that has both translational and rotational degrees of freedom. It represents both membrane and bending actions.

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.

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.

Body Force VectorMechanical loadings within the interior of the volume, typically inertia loadings in a stiffness analysis.

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 ElementA 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 IntegralSee Boundary Element.

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.

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.

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

Cauchy StressSee True Stress.

CellA term used in CFD for a discrete area or volume over which the governing equations are integrated, equivalent to an element in finite element methods. The complete group of cells should define the domain under consideration.

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.

Characteristic ValueSee Eigenvalues.

Characteristic VectorSee Eigenvectors.

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.

Closed-Form Displacement MethodFor fracture mechanics, a special form of displacement substitution that only uses the calculated values in the crack tip elements.

Coefficient Of Viscous DampingThe constants of proportionality relating the velocities to the forces.

Column Vector (Column Matrix)An nx1 matrix written as a vertical string of numbers. It is the transpose of a Row Vector.

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.

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. See also Damped Eigenvalues.

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. See also Damped Eigenvectors.

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 in gaseous fluids where speeds are sufficiently high, causing significant fluid density changes. It typically occurs when the Mach Number exceeds approximately 0.3.

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 NumberThe 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 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.

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.

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 LoadA load that always acts in a fixed direction regardless of the deformation of the body, for example, gravity.

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. See Arc Length Method.

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 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.

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 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.

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.

ConvergenceFor 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 CriterionIn a non-linear solution procedure, this specifies how to decide whether convergence has been achieved within a given increment of time or load.

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.

Coordinate SystemThe set of displacements used to define the degrees of freedom of the system.

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. See also Symmetrical Matrix.

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 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.

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. See also Crack Tip.

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.

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 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.

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.