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# Terms N-R

Natural Frequency
The frequency at which resonance occurs, that is when the stiffness and the inertia forces cancel. See also Resonant Frequency.

Natural Mode
See Eigenvectors.

Natural Strain
See True Strain.

Navier-Stokes Equations
Partial differential equations defining the unsteady viscous flow of fluids.

Newmark Beta Method
An implicit solution method for integrating second order equations of motion. It can be made unconditionally stable. Also referred to as the Newmark Method.

Newmark Method
An implicit solution method for integrating second order equations of motion. It can be made unconditionally stable. Also referred to as the Newmark Beta Method

Newmark’s Time Stepping Schemes
A family of time integration methods for the solution of transient dynamic problems.

Newton Cotes Formulae
A family of methods for numerically integrating a function.

Newton-Raphson Method
An incremental-iterative non-linear procedure to solve the equilibrium equations: the tangential stiffness matrix is updated during every iteration of every increment. See also Newton-Raphson Non-Linear Solution

Newton-Raphson Non-Linear Solution
A general technique for solving non-linear equations. If the function and its derivative are known at any point then the Newton-Raphson method is second order convergent. See also Tangent Stiffness Matrix.

Nodal Values
The value of variables at the node points. For a structure typical possible nodal values are force, displacement, temperature, velocity, x, y, and z.

Nodes
The element behaviour is defined by the response at the nodes of the elements. Nodes are always at the corners of the element, higher order elements have nodes at mid-edge or other edge positions and some elements have nodes on faces or within the element volume. The behaviour of the element is defined by the variables at the node. For a stiffness matrix the variables are the structural displacement, for a heat conduction analysis the nodal variable is the temperature. Other problems have other nodal variables.

Nominal Strain
See Engineering Strain

Non-Associative Plasticity
A form of plasticity in which the yield function is not identical to the plastic potential.

Non-Conforming Elements
Elements that do not satisfy compatibility either within the element or across element boundaries or both. Such elements are not generally reliable although they might give very good solutions in some circumstances.

Non-Holonomic Constraints
Constraints that can only be defined at the level of infinitesimal displacements. They cannot be integrated to give global constraints.

Non-Linear Analysis
When at least one of the coefficients of stiffness, mass or damping vary with displacement or time then the system is non-linear.Superposition cannot be used to solve the problem.

Non-Stationary Random
A force or response that is random and its statistical properties vary with time.

Non-Structural Mass
Mass that is present in the system and will affect the dynamic response but it is not a part of the structural mass (e.g. the payload).

Norm
A scalar measure of the magnitude of a vector or a matrix.

Normality Rule
A particular plastic flow rule to ensure that the plastic strain components are in a ratio such that their resultant is in a direction normal to the yield surface. See also Flow Rule.

Norton-Bailey Law
A law for steady state creep with strain rate proportional to a power of stress. Also referred to as the Norton Law.

Norton Law
A law for steady state creep with strain rate proportional to a power of stress. Also referred to as the Norton-Bailey Law.

Numerical Integration
The process of integrating the element stiffness matrix based on numerical algorithms such as Gaussian quadrature. Evaluations are made at strategic points within each element, known as Gauss Points.

Optimal Points
Strategic locations within elements where stress evaluations are especially accurate, often at the location of the Gauss points.

Optimal Sampling Points
The minimum number of Gauss points required to integrate an element matrix. Also the Gauss points at which the stresses are most accurate, see also Reduced Integration.

Orthotropy
A material where the response to load depends on the direction within the material. It is less general than anisotropy, and up to 12 independent constants are required to relate stress and strain.

Over Damped System
A system that has an equation of motion where the damping is greater than critical. It has an exponentially decaying, non-oscillatory impulse response. See also Critical Damping.

Overstiff Solutions
Lower bound solutions. These are associated with the Assumed Displacement Method.

Parametric Studies
Initial studies conducted on small simplified models to determine the important parameters in the solution of a problem. These are often used to determine the basic mesh density required. Also referred to as Pilot Studies.

Participation Factor
The fraction of the mass that is active for a given mode with a given distribution of dynamic loads. Often this is only defined for a specific load case of inertia (seismic) loads.

Patch Test
A simple element type test using a patch of several elements, one of which is arbitrarily orientated with respect to the global co-ordinates. If the patch is loaded by displacements consistent with a state of constant strain and the strain inside the selected element is constant, the test is passed. See also Continuum Region Element (CRE) Method.

Penalty Function
In the context of contact algorithms, a constraint on stiffness behaviour usually applied via large numbers in the equations, e.g. by introducing stiff springs. Also referred to as Penalty Stiffness.

Penalty Stiffness
In the context of contact algorithms, a constraint on stiffness behaviour usually applied via large numbers in the equations, e.g. by introducing stiff springs. Also referred to as Penalty Function.

Perfect Plasticity
Plastic behaviour where the yield stress remains constant for all values of plastic strain.

Periodic Response (Force)
A response (force) that regularly repeats itself exactly.

Phase Angle
The ratio of the in-phase component of a signal to its out-of-phase component gives the tangent of the phase angle of the signal relative to some reference.

Phase Change
When a substance subjected to temperature changes transforms from solid to liquid or gaseous state. During this phase change, latent heat is either released or absorbed.

Physics Based Modelling/Physics Based Simulation
A subset of “Modelling & Simulation” where the numerical methods that are employed are based on the governing laws of nature.

Pilot Studies
Initial studies conducted on small simplified models to determine the important parameters in the solution of a problem. These are often used to determine the basic mesh density required. Also referred to as Parametric Studies.

Plane Strain
A two dimensional analysis is plane stress if the stress in the third direction is assumed zero. This is valid if the dimension of the body in this direction is very small, e.g. a thin plate. See also Plane Stress.

Plane Stress
A two dimensional analysis is plane strain if the strain in the third direction is assumed zero. This is valid if the dimension of the body in this direction is very large, e.g. a cross-sectional slice of a long body. See also Plane Strain.

Plastic Strain
Irrecoverable permanent strain due to time independent plasticity.

Plastic Zones
Regions in a body where a stress measure (usually the equivalent stress) lies on the yield surface and plastic strains are accruing.

Plate Bending Elements
Two dimensional shell elements where the in plane behaviour of the element is ignored. Only the out of plane bending is considered.

Ply Lay-Up
See Lay-up.

Poissons Ratio
The material property in Hookes law relating strain in one direction arising from a stress in a perpendicular direction to this.

Post Analysis Checks
Checks that can be made on the results after the analysis. For a stress analysis these could include how well stress free boundary conditions have been satisfied or how continuous stresses are across elements.

Post Yield Fracture Mechanics (PYFM)
A given crack inside a loaded structure behaves in conditions of PYFM when the crack fields local to the crack tip exhibit considerable plastic behaviour. See also Elastic Plastic Fracture Mechanics (EPFM).

Post-Processing
The interrogation of the results after the analysis phase. This is usually done graphically.

Potential Energy
The energy associated with the static behaviour of a system. For a structure this is the strain energy.

Potential Energy Release Rate
For a hypothetically small increase in crack length or area, this is the amount of potential energy released divided by that length or area. It equals the negative of the Strain Energy Release Rate when elastic conditions predominate. It provides the basis for fracture parameters in post yield fracture mechanics and other non-linear conditions.

Potential Flow
Fluid flow problems where the flow can be represented by a scalar potential function.

Power Method
A method for finding the lowest or the highest eigenvalue of a system. See also Eigenvalues.

Prandtl Number
A fluid flow measure of the ratio of momentum diffusivity to thermal diffusivity.

Prandtl-Reuss Equations
The equations relating an increment of stress to an increment of plastic strain for a metal undergoing plastic flow. See also Von Mises Yield Criterion.

Prandtl-Reuss Flow Rule
In plasticity theory, the special form of the normality rule corresponding to von Mises yield criterion (q.v.).

Predictor-Corrector Schemes
The two-phase format of a time or load stepping scheme where the predicted solution is corrected prior to advancing to the next step.

P-Refinement
Making the element order higher is termed p-refinement. See also Mesh Refinement

Primary Component
Those parts of the structure that are of direct interest for the analysis. Other parts are Secondary Components.

Primary Creep
The initial part of a creep test where the strain rate is decreasing.

Principal Curvature
The maximum and minimum radii of curvature at a point.

Principal Planes
The planes on which the shear stresses are zero. Three such planes exist at every point in a stressed body.

Principal Stresses
Stresses normal to the principal planes.

Probabilistic Techniques
The purpose of most probabilistic (or stochastic) analyses is to predict the probability of failure. These are normally described as reliability analyses. 

Process Step Item
A Process Step Items acts as the process step record and tracks the execution of software methods that operate on data items in the SDM repository and their attached files.

Product Data Management (PDM)

Product Lifecycle Management (PLM)
Systems that manage a broad set of product information through the lifecycle of the product, project or system being managed.

Product Validation
The confirmation that the Final Product as manufactured does meet all standards, requirements and customer expectations.

Product Verification
The confirmation that the product design has the capability to meet the standards, requirements and design targets demonstrated

Process Integration and Design Optimisation (PIDO)
Solutions that are software applications which enable sequences of simulation process steps to be defined as a workflow and subsequently re-executed.

Professional Simulation Engineer (PSE)
An individual who has achieved NAFEMS Professional Simulation Engineer indicating that they have achieved a certain level of competence in the application of simulation and analysis tools in an industrial setting.

Professional Simulation Engineer Certification
Professional Simulation Engineer Certification allows engineers and analysts within the international simulation community to gain recognition for competencies acquired throughout their professional career. Independently assessed by NAFEMS, the international association of the engineering analysis, modelling and simulation community, the Certification enables individuals to gain recognition for their level of competency and experience as well as enabling industry to identify suitable and qualified personnel. 

Profile
The profile of a symmetric matrix is the sum of the number of terms in the lower (or upper) triangle of the matrix ignoring the leading zeros in each row. Embedded zeros are included in the count. It gives a measure of the work required to factorise the matrix when using the Cholesky solution. It is minimised by node renumbering. See also Cholesky Factorisation (Skyline).

Proportional Damping
A damping matrix that is a linear combination of the mass and stiffness matrices. The eigenvectors of a proportionally damped system are identical to those of the undamped system.

Occurs when all the external loads are applied simultaneously, and increase in proportion to one another throughout the loading history. This clearly does not occur when one component of load is applied and then another.

Qr Method
A technique for finding Eigenvalues This is currently the most stable method for finding eigenvalues but it is restricted in the size of problem that it can solve.

A mode of heat transfer due to electromagnetic waves. Thus, the heat energy can be transferred in a vacuum. It is characterised by the Stefan-Boltzmann law.

Damping that arises from energy being carried away from a vibrating body by expanding pressure waves. Sound radiation is an example of this. Such radiating energy, both to the surrounding fluid and through the supports, often forms the main damping in a vibrating structure.

Ramberg-Osgood Power Law
A stress-strain relationship where the strain is proportional to a power of stress.

Random Vibration Analysis
An analysis undertaken when the applied loading is only known in terms of its statistical properties. The loading is non-deterministic in that its value is not known exactly at any time but its mean, mean square, variance and other statistical quantities are known.

Rank Deficiency
A measure of how singular a matrix is. See Singular Matrix

Ratchetting
Occurs in Cyclic Loading when plastic strains keep on accumulating incrementally with each cycle, leading to eventual failure via incremental collapse.

Rayleigh Damping
A model for representing the variation of damping with frequency.

Raylelgh Quotient
The ratio of stiffness times displacement squared (2 x strain energy) to mass times displacement squared. The minimum values of the Rayleigh quotient are the eigenvalues.

Reaction Forces
The forces generated at support points when a structure is loaded.

Receptance
The ratio of the steady state displacement response to the value of the forcing function for a sinusoidal excitation. It is the same as the Dynamic Flexibility.

Reduced Integration
The process of intentionally under-integrating the element stiffness matrix to prevent problems such as Shear Locking or to improve the element’s performance.

Redundant Structure
A 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.

Requirements Management
The process of documenting, analyzing, tracing, prioritizing and agreeing on requirements and then controlling change and communicating to relevant stakeholders.

Residual Forces
The forces which are equal to the applied load minus the internal resisting forces which occur during non-linear solutions: used to measure the state of equilibrium by comparison to the convergence tolerance.

Resonant Frequency
The frequency at which resonance occurs, that is when the stiffness and the inertia forces cancel. See also Natural Frequency

Resonant Response
The response of a system to vibratory forces.

Response Spectrum Method
A method for characterising a dynamic transient forcing function and the associated solution technique. It is used for seismic and shock type loads.

Restarts Checkpoints
The process whereby an analysis can be stopped part way through and the analysis restarted at a later time.

Reynolds Number
A fluid flow measure of the ratio of momentum forces to viscous forces.

Rigid Body Displacement
A non-zero displacement pattern that has zero strain energy associate with it.

Rigid Body Modes
If a displaced shape does not give rise to any strain energy in the structure then this a rigid body mode. A general three dimensional unsupported structure has 6 rigid body modes, 3 translation and 3 rotation.

This is a connection between two non-coincident nodes assuming that the connection is infinitely stiff. This allows the degrees of freedom at one of the nodes (the slave node) to be deleted from the system. It is a form of Multi-Point Constraints.

Rigid Offsets
This is a connection between two non-coincident nodes assuming that the connection is infinitely stiff. This allows the degrees of freedom at one of the nodes (the slave node) to be deleted from the system. It is a form of Multi-Point Constraints.

Robustness
Some designs are inherently less sensitive to intrinsic variability (noise). This is often a design goal and needs to be demonstrated. Design for Six Sigma, for example, prioritises the Robustness of the product, measured in terms of its failure rate, overachieving maximum performance. 

Roundoff Error
Computers have a fixed wordlength and hence only hold numbers to a certain number of significant figures. If two close numbers are subtracted one from another then the result loses the first set of significant figures and hence loses accuracy. This is round off error.

Row Vector
A 1xn matrix written as a horizontal string of numbers. It is the transpose of a Column Vector (Column Matrix)Column Vector (Column Matrix)

Rupture Time
The time required for a structure to fail due to continuous creep deformation.

 NAFEMS, Professional Simulation Engineer Certification. [online] Available at: https://www.nafems.org/professional-development/certification/.

 NAFEMS. (2012). What is a Probabilistic or Stochastic Analysis?. [online] Available at: https://www.nafems.org/publications/resource_center/wt01/ [Accessed 24 May 2022]

 Bartholomew, P. (2014). What is sensitivity analysis? NAFEMS. [online] Available at: https://www.nafems.org/publications/resource_center/wt04/ [Accessed 24 May 2022]