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## Finite Element Analysis FEA Terms and Definitions (A to Z) Part-6 (Final)

Finite Element Analysis FEA Terms and Definitions (A to Z) Part-5

### T

TEMPERATURE CONTOUR PLOTS
A plot showing contour lines connecting points of equal temperature.
TETRAHEDRON TETRAHEDRAL ELEMENT
A three dimensional four sided solid element with triangular faces.
THERMAL CAPACITY
The material property defining the thermal inertia of a material. It relates the rate of change of temperature with time to heat flux.
THERMAL CONDUCTIVITY
The material property relating temperature gradient to heat flux.
The equivalent loads on a structure arising from thermal strains. These in turn arise from a temperature change.
THERMAL STRAINS
The components of strain arising from a change in temperature.
THERMAL STRESS ANALYSIS
The computation of stresses and displacements due to change in temperature.
THIN SHELL ELEMENT THICK SHELL ELEMENT
In 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 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.
TIMEDOMAIN
The 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.
TRACE OF THE MATRIX
The sum of the leading diagonal terms of the matrix.
TRANSFINITE MAPPING
A systematic method for generating element shape functions for irregular node distributions on an element.
TRANSFORMATION METHOD
Solution 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 FORCE
A forcing function that varies for a short period of time and then settles to a constant value.
TRANSIENT RESPONSE
The component of the system response that does not repeat itself regularly with time.
TRANSITION ELEMENT
Special 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.
TRANSIENT HEAT TRANSFER
Heat transfer problems in which temperature distribution varies as a function of time.
TRIANGULAR ELEMENTS
Two dimensional or surface elements that have three edges.
TRUSS ELEMENT
A one dimensional line element defined by two nodes resisting only axial loads.

### U

ULTIMATE STRESS
The failure stress (or equivalent stress) for the material.
UNDAMPED NATURAL FREQUENCY
The square root of the ratio of the stiffness to the mass (the square root of the eigenvalue). It is the frequency at which an undamped system vibrates naturally. A system with n degrees of freedom has n natural frequencies.
UNDER DAMPED SYSTEM
A system which has an equation of motion where the damping is less than critical. It has an oscillatory impulse response.
UNIT MATRIX
A diagonal matrix with unit values down the diagonal.
UPDATED LAGRANGIAN TOTAL LAGRANGIAN
The updated Lagrangian coordinate system is one where the stress directions are referred to the last known equilibrium state. The total Lagrangian coordinate system is one where the stress directions are referred to the initial geometry.
UPWINDING IN FLUIDS
A special form of weighting function used in viscous flow problems (solution to the NavierStokes equations) used in the weighted residual method to bias the results in the direction of the flow.

### V

VARIABLE BANDWIDTH (SKYLINE)
A sparse matrix where the bandwidth is not constant. Some times called a skyline matrix.
VELOCITY
The first time derivative of the displacement.
VIRTUAL CRACK EXTENSION CRACK PROPAGATION
A technique for calculating the energy that would be released if a crack increased in size. This gives the energy release rate which can be compared to the critical energy release (a material property) to decide if a crack will propagate.
VIRTUAL DISPLACEMENTS
An arbitrary imaginary change of the system configuration consistent with its constraints.
VIRTUAL WORK VIRTUAL DISPLACEMENTS VIRTUAL FORCES
Techniques for using work arguments to establish equilibrium equations from compatibility equations (virtual displacements) and to establish compatibility equations from equilibrium (virtual forces).
VISCOUS DAMPING
The damping is viscous when the damping force is proportional to the velocity.
VISCOUS DAMPING MATRIX
The matrix relating a set of velocities to their corresponding velocities
VOLUME DISTORTION VOLUMETRIC DISTORTION
The distortion measured by the determinant of the Jacobian matrix, det j.
VON MISES STRESS
An “averaged” stress value calculated by adding the squares of the 3 component stresses (X, Y and Z directions) and taking the square root of their sums. This value allows for a quick method to locate probable problem areas with one plot.
VON MISES EQUIVALENT STRESS TRESCA EQUIVALENT STRESS
Equivalent stress measures to represent the maximum shear stress in a material. These are used to characterize flow failures (e.g. plasticity and creep). From test results the VonMises form seems more accurate but the Tresca form is easier to handle.

### W

WAVE PROPAGATION
The 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 minimized be element renumbering.
WEIGHTED RESIDUALS
A 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 process. This leads to identical equations to those from virtual work arguments.
WHIRLING STABILITY
The stability of rotating systems where centrifugal and Coriolis are also present.
WHITE NOISE
White noise has a constant spectral density for all frequencies.
WILSON THETA METHOD
An implicit solution method for integrating second order equations of motion. It can be made unconditionally stable.
WORD LENGTH
Within 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.

### Y

YOUNG’S MODULUS
The material property relating a uniaxial stress to the corresponding strain.

### Z

ZERO ENERGY MODES ZERO STIFFNESS MODES
Non-zero patterns of displacements that have no energy associated with them. No forces are required to generate such modes, Rigid body motions are zero energy modes. Buckling modes at their buckling loads are zero energy modes. If the elements are not fully integrated they will have zero energy displacement modes. If a structure has one or more zero energy modes then the matrix is singular.

###### REFERENCES

http://www.cae.tntech.edu/~chriswilson/FEA/ANSYS/

www.nafems.org

## Finite Element Analysis FEA Terms and Definitions (A to Z) Part-5

Finite Element Analysis FEA Terms and Definitions (A to Z) Part-4

### O

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 reduced Gauss points).
OVER DAMPED SYSTEM
A system which has an equation of motion where the damping is greater than critical. It has an exponentially decaying, non-oscillatory impulse response.
OVERSTIFF SOLUTIONS
Lower bound solutions. These are associated with the assumed displacement method.

### P

PARAMETRIC STUDIES 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.
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 the specific load case of inertia (seismic) loads.

PATCH TEST
A test to prove that a mesh of distorted elements can represent constant stress situations and strain free rigid body motions (i.e. the mesh convergence requirements) exactly.
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.
PLANE STRAIN PLANE STRESS
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. 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.
PLATE BENDING ELEMENTS
Two-dimensional shell elements where the in plane behavior of the element is ignored. Only the out of plane bending is considered.
POISSONS RATIO
The material property in Hooke s 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-PROCESSING
The interrogation of the results after the analysis phase. This is usually done with a combination of graphics and numerics.

https://www.udemy.com/ansys-designmodeler-essentials/?couponCode=DESIGNMODELER

POTENTIAL ENERGY
The energy associated with the static behavior of a system. For a structure this is the strain energy.
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.
PRANDTL-REUSS EQUATIONS
The equations relating an increment of stress to an increment of plastic strain for a metal undergoing plastic flow.
PREPROCESSING
The process of preparing finite element input data involving model creation, mesh generation,material definition, and load and boundary condition application.
PRIMARY COMPONENT

Those parts of the structure that are of direct interest for the analysis. Other parts are secondary components.
PRINCIPAL CURVATURE
The maximum and minimum radii of curvature at a point.
PRINCIPAL STRESSES
The maximum direct stress values at a point. They are the eigenvalues of the stress tensor.
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 factorize the matrix when using the Cholesky solution. Node renumbering minimizes it.
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.
P-METHOD
A method of finite element analysis that uses P- convergence to iteratively minimize the error of analysis.
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.
RANDOM VIBRATIONS
The applied loading is only known in terms of its statistical properties. The loading is nondeterministic in that its value is not known exactly at any time but its mean, mean square, variance and other statistical quantities are known.

### R

RANK DEFICIENCY
A measure of how singular a matrix is.
RAYLEIGH DAMPING
Damping that is proportional to a linear combination of the stiffness and mass. This assumption has no physical basis but it is mathematically convenient to approximate low damping in this way when exact damping values are not known.
RAYLELGH QUOTIENT
The ratio of stiffness times displacement squared (2*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.
REFERENCE TEMPERATURE
The reference temperature defines the temperature at which strain in the design does not
result from thermal expansion or contraction. For many situations, reference temperature
is adequately defined as room temperature. Define reference te mperature in the
properties of an environment.
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
If an element requires an l*m*n Gauss rule to integrate the element matrix exactly then (l-1)*(m-1)*(n-1) is the reduced integration rule. For many elements the stresses are most
accurate at the reduced integration points. For some elements the matrices are best
evaluated by use of the reduced integration points. Use of reduced integration for
integrating the elements can lead to zero energy and hour glassing modes.
RESPONSE SPECTRUM METHOD
A method for characterizing 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.
RIGID BODY DEFORMATIONS
A non-zero displacement pattern that h as zero strain energy associate with it.
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 constraint.
ROUNDOFF ERROR
Computers have a fixed word length 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 ROW MATRIX
A 1xn matrix written as a horizontal string of numbers. It is the transpose of a column vector.

### S

SCALARS VECTORS
Quantities that have no direction associated with them, e.g. temperatures. Scalar problems only have one degree of freedom at a node. Vector quantities have a direction associated with them, e.g. displacements. Vector problems have more than one degree of freedom at a node.
SECANT STIFFNESS
The stiffness defined by the slope of the line from the origin to the current point of interest on a load/deflection curve.
SECONDARY COMPONENTS

Components of a structure not of direct interest but they may have some influence of the behavior 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.
SEEPAGE FLOW
Flows in porous materials
SEISMIC ANALYSIS
The calculation of the dynamic displacement and stress response arising from earthquake excitations.
SELECTED REDUCED INTEGRATION
A form of Gaussian quadrature where different sets of Gauss points are used for different strain components.
A form of matrix products that preserves symmetry of equations. The product A*B*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 o f 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.
A load set is self -equilibrating if all of its resultants are zero. Both translation and moment resultants are zero.
SEMI-LOOF ELEMENT
A form of thick shell element.
SHAKEDOWN
If a structure is loaded cyclically and initially undergoes some plastic deformation then it is said to shakedown if the behavior is entirely elastic after a small number of load cycles.
SIMPSONS RULE
A method for numerically integrating a function.
SIMULTANEOUS VECTOR ITERATION
A 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 FREEDOM
The system is defined by a single force/displacement equation.
SINGLE POINT CONSTRAINT
Where the constraint is unique to a single node point.
SINGULAR MATRIX
A 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.
SOLID ELEMENTS

Three dimensional continuum elements.
SOLUTION DIAGNOSTICS
Messages that are generated as the finite element solution progresses. These should always be checked for relevance but the are often only provided for information purposes
SOLUTION EFFICIENCY
A comparative measure between two solutions of a given problem defining which is the ‘best’. The measures can include accuracy, time of solution, memory requirements and disc storage space.
SPARSE MATRIX METHODS
Solution methods that exploit the sparse nature of finite element equations. Such methods include the frontal solution and Cholesky (skyline) factorization for direct solutions, conjugate gradient methods for iterative solutions and the Lanczos method and subspace iteration (simultaneous vector iteration) for eigenvalue solutions.
SPECTRAL DENSITY
The 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 CURVES
A curve fitting technique that preserves zero, first and second derivative continuity across segment boundaries.
SPURIOUS CRACKS
Cracks that appear in a mesh when the elements are not correctly connected together. This is usually an error in the mesh generation process.
STATIC ANALYSIS
Analysis of stresses and displacements in a structure when the applied loads do not vary with time.
STATICALLY DETERMINATE STRUCTURE
A structure where all of the unknowns can be found from equilibrium considerations alone.
Equivalent 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 INDETERMINATE STRUCTURE REDUNDANT
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.
STATIONARY RANDOM EXCITATION
A force or response that is random but its statistical characteristics do not vary with time.
Determination of the temperature distribution of a mechanical part having reached thermal equilibrium with the environmental conditions. There are no time varying changes in the resulting temperatures.

The response of the system to a periodic forcing function when all of the transient components of the response have become insignificant.
STEP-BY-STEP INTEGRATION
Methods of numerically integrating time varying equations of motion. These methods can be either explicit or implicit.
STIFFNESS
A set of values which represent the rigidity or softness of a particular element. Stiffness is determined by material type and geometry.
STIFFNESS MATRIX
The 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
A dimensionless quantity calculated as the ratio of deformation to the original size of the body.
STRAIN ENERGY
The energy stored in the system by the stiffness when it is displaced from its equilibrium position.
STRESS
The intensity of internal forces in a body (force per unit area) acting on a plane within the material of the body is called the stress on that plane.
STRESS ANALYSIS
The computation of stresses and displacements due to applied loads. The analysis may be elastic, inelastic, time dependent or dynamic.
STRESS AVERAGING STRESS SMOOTHING
The process of filtering the raw finite element stress results to obtain the most realistic estimates of the true state of stress.
STRESS CONCENTRATION
A 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 CONTOUR PLOT
A plot of a stress component by a series of color filled contours representing regions of equal stress.
STRESS DISCONTINUITIES STRESS ERROR ESTIMATES
Lines along which the stresses are discontinuous. If the geometry or loading changes abruptly along a line then the true stress can be discontinuous. In 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.
STRESS EXTRAPOLATION
The process of taking the stress results at the optimum sampling points for an element and extrapolating these to the element node points.
STRESS INTENSITY FACTORS

A measure of the importance of the stress at a sharp crack tip (where the actual stress values will be infinite) used to estimate if the crack will propagate.

STRESS VECTOR STRESS TENSOR STRAIN VECTOR STRAIN TENSOR
The stress (strain) vector is the components of stress (strain) written as a column vector. For a general three dimensional body this is a (6×1) matrix. The components of stress (strain) written in tensor form. For a general three dimensional body this forms a (3×3) matrix with the direct terms down the diagonal and the shear terms as the off-diagonals.
STRESS-STRAIN LAW
The material property behavior relating stress to strain. For a linear behavior this is Hookes law (linear elasticity). For elastic plastic behavior it is a combination of Hookes law and the Prandtl-Reuss equations.
SUBSPACE VECTOR ITERATION
A method for finding the first few eigenvalues and eigenvectors of a finite element system. This is also known as simultaneous vector iteration.
SUBSTRUCTURING
An efficient way of solving large finite element analysis problems by breaking the model into several parts or substructures, analyzing each one individually, and then combining them for the final results.
SUBSTRUCTURING SUPER ELEMENT METHOD
Substructuring 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 gives the super element matrix
for the substructure.
SURFACE MODELING
The geometric modeling technique in which the model is created in terms of its surfaces only without any volume definition.

Finite Element Analysis FEA Terms and Definitions (A to Z) Part-6 (Final)

###### REFERENCES

http://www.cae.tntech.edu/~chriswilson/FEA/ANSYS/

www.nafems.org

## Revit MEP- State of the Art BIM Software

### Introduction

Autodesk Revit MEP is a building information modeling (BIM) software created by Autodesk for professionals who engage in MEP engineering. MEP stands for mechanical, electrical, and plumbing, which are the three engineering disciplines that Revit MEP addresses. By utilizing BIM as opposed to computer-aided drafting (CAD), Revit MEP is able to leverage dynamic information in intelligent models — allowing complex building systems to be accurately designed and documented in a shorter amount of time. Each intelligent model created with Revit MEP represents an entire project and is stored in a single database file. This allows changes made in one part of the model to be automatically propagated to other parts of the model, thus enhancing the workflow for Revit MEP users.

[Source: EduLearn]

Autodesk® Revit® MEP software helps mechanical, electrical, and plumbing engineering firms meet the heightened demands of today’s global marketplace. [Autodesk Revit MEP
facilitated collaboration
among all the teams
on a single, fully
coordinated parametric
model, enabling us
to deliver integrated
solutions that bypassed
the problems inherent
in drawing-based
technologies.
—Stanis Smith
Senior Vice President
Stantec]

#### BIM for Mechanical, Electrical, and Plumbing Engineers

Autodesk® Revit® MEP software is the building information modeling (BIM) solution for mechanical, electrical, and plumbing (MEP) engineers, providing purpose-built tools for building systems design and analysis. With
Revit MEP, engineers can make better decisions earlier in the design process because they can accurately visualize building systems before they are built. The software’s built-in analysis capabilities helps users create more sustainable designs and share designs using a wide variety of partner applications, resulting in optimal building performance and efficiency. Working with a building information model helps keep design data coordinated, minimizes errors, and enhances collaboration among engineering and architecture teams.

#### Building Systems Modeling and Layout

Revit MEP software’s modeling and layout tools enable engineers to create mechanical, electrical, and plumbing systems more accurately and easily. Automatic routing solutions enable users to model the ductwork, plumbing, and piping systems, or manually lay out lighting and power systems. Revit MEP software’s parametric change technology
means that any change to the MEP model is automatically coordinated throughout the model. Maintaining a single, consistent model of the building helps to keep drawings coordinated and reduce errors.

### 1. Duct and Pipe System Modeling

Intuitive layout tools enable easier model modifications. Revit MEP automatically updates model views and sheets, helping to maintain document and project consistency. Engineers can create HVAC systems with mechanical
functionality and provide 3D modeling for ductwork and piping as well as modify the model by dragging design elements onto the screen in almost any view. Modeling can also be done in both section and elevation views. All model views and sheets update automatically whenever a change is made anywhere for more accurate, coordinated designs and documents.

#### 2. Duct and Pipe Sizing/Pressure Calculations

With built-in calculators in Autodesk Revit MEP software, engineers can perform sizing and pressure loss calculations according to industrystandard methods and specifications, including the American Society of Heating, Refrigerating, and Air-Conditioning Engineers (ASHRAE) fitting loss database. System sizing tools instantly update
the size and design parameters of duct and pipe elements without the need for file exchanges or third-party applications. Select a dynamic sizing method for the ductwork and piping systems in your plans using duct sizing and pipe sizing tools, including friction, velocity, static regain, and equal friction sizing method for duct sizing, and velocity or friction method for pipe sizing. #### 3. HVAC and Electrical System Design

Communicate design intent visually with room color-fill plans. With color schemes, team members no longer have to spend time deciphering spreadsheets and using colored pencils on printed plans. All revisions and alterations to color-fill plans are updated automatically across the model. Create any number of schemes, and maintain better
consistency for the duration of the project. Three dimensional modeling for ductwork and piping enables users to create HVAC systems that can be clearly shown using color schemes for design airflow, actual airflow, mechanical zones, and more. Create electrical color schemes for power loads, lighting per area, and more. #### 4. Conduit and Cable Tray Modeling

Revit MEP contains powerful layout tools that enable easier modeling of electrical and data cable trays and conduit. Better coordinate and create accurate construction drawings using real-world conduit and cable tray combinations.
New schedule types can report the overall length of cable tray and conduit runs, resulting in rapid quantification of required materials. #### 5. Automatic Generation of Construction Document Views

Automatically generate plan, section, elevation detail, and schedule views that more precisely reflect design information. Synchronized model views from a common database enable more consistent, coordinated change management. The entire electrical, plumbing, and mechanical design team benefits from more accurate, coordinated construction documents that building information modeling provides. Leverage the millions of professionally trained AutoCAD users worldwide to share and complete MEP projects faster. Revit MEP provides seamless support for AutoCAD software’s DWG™ file format enabling you to save and share files with confidence. DWG technology from Autodesk is the authentic, accurate, and reliable way to store and share design data. References

https://www.edulearn.com/article/what_is_revit_mep.html

## Formula 1 Cars Evolution, Design and Components

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Fundamentally, Formula One cars are no different than the Chevy parked out in your garage. They use internal combustion engines and have transmissions, suspensions, wheels and brakes. But that’s where the similarity ends. Formula One cars aren’t designed for casual driving or cruising down the interstate. Everything about them is tweaked and tooled for one thing and one thing only — speed. Formula One cars can easily attain speeds of 200 mph — but during a race, the speeds are generally lower. During the 2006 Hungarian Grand Prix, the winner’s average speed was 101.769 mph, and in the 2006 Italian Grand Prix, it was 152.749 mph.

A Formula One car is an open-wheel, open-cockpit, single-seat racing car for the purpose of being used in Formula One competitions. It is equipped with two wings (front and rear) plus an engine, which is located behind the driver. #### Construction

Every F1 car is composed of two main components − the chassis and the engine.

Chassis − Formula One cars these days are made from carbon fiber and ultra-lightweight components. The weight must be not less than 702 kg or 1548 lbs, including the driver and tires, but excluding the fuel.

The dimensions of a Formula One car must be maximum 180 cm (width) × 95cm (height); there is no specified number for maximum length, but all cars tend to be of almost the same length.

The heart of a Formula One car is the chassis — the part of the automobile onto which everything is bolted and attached. Like most modern cars and aircraft, Formula One race cars feature Monocoque construction. Monocoque is a French word meaning “single shell,” which refers to the process of making the entire body out of a single piece of material. Once upon a time, that material was aluminum, but today it’s a strong composite, like spun carbon fibers set in resin or carbon fiber layered over aluminum mesh. The result is a lightweight car that can withstand the enormous downward-acting forces that are produced as the vehicle moves through the air.

The Monocoque incorporates the cockpit, a strong, padded cell that accommodates a single driver. Unlike the cockpits of road-ready cars, which can show great variance, the cockpits of Formula One cars must adhere to very rigorous technical regulations. They must, for example, meet minimum size requirements and must have a flat floor. The seat, however, is made to fit a driver’s precise measurements so his movement is limited as the car moves around the track.

Engine − According to regulation changes in 2014, all F1 cars must deploy 1.6 liter turbocharged V6 engines. Before 2006, Formula One cars were powered by massive three-liter, V10 engines. Then the rules changed, specifying the use of 2.4-liter V8 engines. Even though power outputs fell with the rule change, Formula One engines­ can still produce nearly 900 horsepower. To put that into perspective, consider that a Volkswagen Jetta’s 2.5-liter engine produces just 150 horsepower. Of course, the Jetta’s engine is probably good for at least 100,000 miles or so. A Formula One engine needs to be rebuilt after about 500 miles. Why? Because generating all of that power requires that the engine run at very high revolution rates — nearly 19,000 revolutions per minute. Running an engine at such high rpms produces an enormous amount of heat and puts a great deal of stress on the moving parts.

#### Formula One Transmissions and Aerodynamics

##### Transmission(Gearbox)

Semi-automatic sequential carbon titanium gearboxes are used by F1 cars presently, with 8 forward gears and 1 reverse gear, with rear-wheel drive. It’s the job of the transmission to transfer all of the engine’s power to the rear wheels of the Formula One car. The transmission bolts directly to the back of the engine and includes all of the parts you would expect to find in a road car — gearbox, differential and driveshaft. The gearbox must have a minimum of four forward gears and a maximum of seven gears. Six-speed gearboxes were popular for several years, but most Formula One cars now run seven-speed units. A reverse gear must also be fitted. The gearbox is connected to a differential, a set of gears allowing the rear wheels to revolve at different speeds during cornering. And the differential is connected to the driveshaft, which transfers power to the wheels.­ Shifting gears in a Formula One car is not the same as shifting gears in a road car with a manual transmission. Instead of using a traditional “H” gate selector, drivers select gears using paddles located just behind the steering wheel. Downshifting is done on one side of the steering wheel, upshifting on the other. Although fully automatic transmission systems, including systems with sophisticated launch control, are possible on Formula One cars, they are now illegal. This helps reduce the overall cost of the power train and enables drivers to use gear-shifting skills to gain advantage in a race.

##### Aerodynamics

A Formula One race car is defined as much by its aerodynamics as it is by its powerful engine. That’s because any vehicle traveling at high speed must be able to do two things well: reduce air resistance and increase downforce. Formula One cars are low and wide to decrease air resistance. Wings, a diffuser, end plates and barge boards increase downforce. Let’s look at each of these in greater detail.

• Wings, which first appeared in the 1960s, operate on the same principles as airplane wings, only in reverse. Airplane wings create lift, but the wings on a Formula One car produce downforce, which holds the car onto the track, especially during cornering. The angle of both front and rear wings can be fine-tuned and adjusted to get the ideal balance between air resistance and downforce.
• Lotus engineers discovered in the 1970s that a Formula One car itself could be turned into a giant wing. Using a unique undercarriage design, they were able to extract air from beneath the car, creating an area of low pressure that sucked the entire vehicle downward. These so-called “ground-effect” forces were soon outlawed and strict regulations put in place to govern undercarriage design. The bottom of today’s cars must be flat from the nose cone to the rear axle line. Beyond that line, engineers have free reign. Most incorporate a diffuser, an upward-sweeping device located just beneath the engine and gearbox that creates a suction effect as it funnels air up and passes it to the rear of the car.
• Much of aerodynamics is concerned with getting air to move where you want it to move. Endplates are small, flanged areas at the edges of the front wings that help “grab” the air and direct it along the side of the car. The barge boards, located just behind the front wheels, pick up the air from there, accelerating it to create even more downforce.

The result of all this aerodynamics engineering is a combined downforce of about 2,500 kilograms (5,512 pounds). That’s more than four times the weight of the car itself.

#### Suspension

he suspension of a Formula One car has all o­f the same components as the suspension of a road car. Those components include springs, dampers, arms and anti-sway bars. These cars feature multi-link suspensions, which use a multi-rod mechanism equivalent to a double-wishbone system. a double-wishbone design uses two wishbone-shaped control arms to guide each wheel’s up-and-down motion. Each arm has three mounting positions — two at the frame and one at the wheel hub — and each joint is hinged to guide the wheel’s motion. In all cars, the primary benefit of a double-wishbone suspension is control. The geometry of the arms and the elasticity of the joints give engineers ultimate control over the angle of the wheel and other vehicle dynamics, such as lift, squat and dive. Unlike road cars, however, the shock absorbers and coil springs of a Formula One race car don’t mount directly to the control arms. Instead, they are oriented along the length of the car and are controlled remotely through a series of pushrods and bell cranks. In such an arrangement, the pushrods and bell cranks translate the up-and-down motions of the wheel to the back-and-forth movement of the spring-and-damper apparatus.

#### Steering Wheel

The steering wheel of an F1 car is equipped to perform many functions like changing gears, changing brake pressure, calling the radio, fuel adjustment, and so on. The steering wheel of a Formula One car bears little resemblance to the steering wheel of a road car. As the car’s command center, it houses a dizzying array of buttons, toggles and switches. During the race, the driver can control almost every aspect of the car’s performance — gear changes, fuel mixture, brake balance and more — with just the touch of a finger. And, amazingly, all of this control comes on a steering wheel that is about half the diameter of a normal car’s steering wheel.

The rules state that the driver must be able to get out of his car within five seconds, removing nothing except the steering wheel. To allow for this, the steering wheel is joined to the steering column via a snap-on connector.

#### Fuel

The fuel used by Formula One cars is a tightly controlled mixture of ordinary petrol, and can only contain commercial gasoline compounds rather than alcohol compounds. The fuel is not the typical unleaded gasoline you pump at the neighborhood Exxon, but it’s similar. Small quantities of non-hydrocarbon compounds are allowed, but most power-boosting additives have been banned completely. All in all, Formula One teams use about 50 different fuel blends, tuned for different tracks or conditions, in a typical season. Each blend must be submitted to the FIA, the sport’s governing body, for approval of its composition and physical properties.

#### Tires

Formula One cars have been using smooth thread, slick tires since 2009. The tire dimensions of an F1 car are −

• Front Tire − 245mm (width)
• Rear Tires − 355mm and 380mm (width)

The tires of a Formula One race car may be the most important part on the entire vehicle. This seems like an overstatement until you realize that the tires are the only things touching the track surface. That means all of the other major systems — engine, suspension and braking — do their work by way of the tires. If the tires don’t perform well, the car won’t perform well, regardless of the technical superiority demonstrated in other systems.

Like every part of a Formula One car, tires are highly regulated. Slick tires — those with no tread pattern and a high contact area — were introduced in the 1960s and used until 1998. Then the FIA change the rules to reduce cornering speeds and make the sport more competitive. On today’s Formula One cars, the front tires must be between 12 and 15 inches wide and the rear tires between 14 and 15 inches wide. Four continuous, longitudinal grooves must run around the circumference. The grooves must be at least 2.5 millimeters (0.098 inches) deep and 50 mm (1.97 inches) apart. In rainy conditions, cars can have “intermediate” and “wet” tires, which have full tread patterns designed to channel water away from the road surface.

Formula One tires are made from very soft rubber compounds which, as they heat up, adhere to the road and provide enormous gripping power. In fact, racing tires perform best at high temperatures, so they have to be warmed up before they are race-ready. The tradeoff is decreased durability. A Formula One tire is designed to last for, at most, about 125 miles.

Traction control can extend the life of tires by limiting wheel spin, especially under loads imposed by cornering. Traction control systems use electronic sensors to compare the speed of the wheel to the speed of the road the wheel is driving over. If the wheel is traveling faster than the road surface — an indication that the wheels are dangerously close to spinning — then the engine is automatically throttled back. Traction control has been allowed and banned at various times throughout modern Formula One history.

#### Brakes

Formula One cars use disc brakes with a rotor and caliper at each tire. You would recognize all of the parts of the disc brakes found on Formula One cars. The big difference, of course, is that the brakes used in Formula One must stop a vehicle traveling at speeds greater than 200 mph. This causes the brakes to glow red-hot when they are used. To help reduce wear and tear and increase braking performance, carbon fiber discs and pads are now used. These brake systems are extremely effective at temperatures up to 750° C (1,382° F), even though they are lightweight. Holes around the edge of the brake disc allow heat to escape rapidly. The cars also have air intakes fitted to the outside of the wheel hub to cool down the brakes. The air intakes are changed for the different braking requirements of each track.

#### Speed and Performance

All F1 cars can accelerate from 0 to 100 mph (160 kmph) and decelerate back to 0 in under 5 seconds. F1 cars have reached top speeds of about 300 kmph or 185 mph on an average.

However, some cars, without fully complying with F1 standards have attained speed of 400 km/h or more. These numbers are mostly same for all F1 cars but slight variations may be there due to the gears and aerodynamics configuration.

###### References
1. https://www.wired.com/2017/03/formula-one-f1-race-carevolution/
2. https://www.tutorialspoint.com/formula_one/car_design_specs_rules.htm
3. http://auto.howstuffworks.com/auto-racing/motorsports/formula-one2.htm

## Finite Element Analysis FEA Terms and Definitions (A to Z) Part-4

##### (A to Z) of Finite Element Analysis

Finite Element Analysis FEA Terms and Definitions (A to Z) Part-3

### L

LAGRANGE INTERPOLATION LAGRANGE SHAPE FUNCTIONS
A method of interpolation over a volume by means of simple polynomials. This is the basis of most of the shape function definitions for elements.
LAGRANGE MULTIPLIER TECHNIQUE
A method for introducing constraints into an analysis where the effects of the constraint are represented in terms of the unknown Lagrange multiplying factors.
LANCZOS METHOD
A method for finding the first few eigenvalues and eigenvectors of a set of equations. It is very well suited to the form of equations generated by the finite element method. It is closely related to the method of conjugate gradients used for solving simultaneous equations iteratively.
LEAST SQUARES FIT
Minimization of the sum of the squares of the distances between a set of sample points and a smooth surface . The finite element method gives a solution that is a least squares fit to the equilibrium equations.
LINEAR DEPENDENCE
One or more rows (columns) of a matrix are linear combinations of the other rows (columns). This means that the matrix is singular.
LINEAR ANALYSIS
Analysis in which the displacements of the structure are linear functions of the applied loads.
LINEAR SYSTEM
When the coefficients of stiffness, mass and damping are all constant then the system is linear. Superposition can be used to solve the response equation.
The loads applied to a structure that result in deflections and consequent strains and stresses.
LOCAL STRESSES
Areas of stress that are significantly different from (usually higher than) the general stress level.
LOWER BOUND SOLUTION UPPER BOUND SOLUTION
The assumed displacement form of the finite element solution gives a lower bound on the maximum displacements and strain energy (i.e. these are under estimated) for a given set of forces. This is the usual form of the finite element method. The 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.
LUMPED MASS MODEL
When the coefficients of the mass matrix are combined to produce a diagonal matrix. The total mass and the position of the structures center of gravity are preserved.

### M

MASS

The constant(s) of proportionality relating the acceleration(s) to the force(s). For a discrete parameter multi degree of freedom model this is usually given as a mass matrix.
MASS ELEMENT
An element lumped at a node representing the effect of a concentrated mass in different coordinate directions.
MASS MATRIX
The matrix relating acceleration to forces in a dynamic analysis. This can often be approximated as a diagonal matrix with no significant loss of accuracy.
MASTER FREEDOMS
The freedoms chosen to control the structural response when using a Guyan reduction or sub structuring methods.
MATERIAL LOSS FACTOR
A measure of the damping inherent within a material when it is dynamically loaded.
MATERIAL PROPERTIES
The physical properties required to define the material behavior for analysis purposes. For stress analysis typical required material properties are Young’s modulus, Poisson’s ratio, density and coefficient of linear expansion. The material properties must have been obtained by experiment.
MATERIAL STIFFNESS MATRIX MATERIAL FLEXIBILITY MATRIX
The material stiffness matrix allows the stresses to be found from a given set of strains at a point. The material flexibility is the inverse of this, allowing the strains to be found from a given set of stresses. Both of these matrices must be symmetric and positive definite.
MATRIX DISPLACEMENT METHOD
A form (the standard form) of the finite element method where displacements are assumed over the element. This gives a lower bound solution.
MATRIX FORCE METHOD
A form of the finite element method where stresses (internal forces) are assumed over the element. This gives an upper bound solution.
MATRIX INVERSE
If matrix A times matrix B gives the unit matrix then A is the inverse of B (B is the inverse of A). A matrix has no inverse if it is singular.
MATRIX NOTATION MATRIX ALGEBRA
A form of notation for writing sets of equations in a compact manner. Matrix notation highlights the generality of various classes of problem formulation and solution. Matrix algebra can be easily programmed on a digital computer.
MATRIX PRODUCTS
Two matrices A and B can be multiplied together if A is of size (j*k) and B is of size (k*l). The resulting matrix is of size (j*l).
MATRIX TRANSPOSE
The process of interchanging rows and columns of a matrix so that the j’th column becomes the j’th row.
MEAN SQUARE CONVERGENCE

A measure of the rate of convergence of a solution process. A mean square convergence indicates a rapid rate of convergence.
MEMBRANE
Membrane behavior is where the strains are constant from the center line of a beam or center surface of a plate or shell. Plane sections are assumed to remain plane. A membrane line element only has stiffness along the line, it has zero stiffness normal to the line. A membrane plate has zero stiffness normal to the plate. This can cause zero energy (no force required) displacements in these normal directions. If the stresses vary linearly along the normal to the centerline then this is called bending behavior.
MESH DENSITY MESH REFINEMENT
The mesh density indicates the size of the elements in relation to the size of the body being analyzed. The mesh density need not be uniform all over the body There can be areas of mesh refinement (more dense meshes) in some parts of the body. Making the mesh finer is generally referred to as h -refinement. Making the element order higher is referred to as p -refinement.
MESH GENERATION ELEMENT GENERATION
The process of generating a mesh of elements over the structure. This is normally done automatically or semi-automatically.
MESH SPECIFICATION
The process of choosing and specifying a suitable mesh of elements for an analysis.
MESH SUITABILITY
The appropriate choice of element types and mesh density to give a solution to the required degree of accuracy.
MINDLIN ELEMENTS
A form of thick shell element.
MOBILITY
The ratio of the steady state velocity response to the value of the forcing function for a sinusoidal excitation.
MODAL DAMPING
The damping associated with the generalized displacements defined by the eigenvectors. Its value has no physical significance since the eigenvector contains an arbitrary normalizing factor.
MODAL MASS
The mass associated with the generalized displacements defined by the eigenvectors. Its value has no physical significance since the eigenvector contains an arbitrary normalizing factor but the ratio of modal stiffness to modal mass is always the eigenvalue.
MODAL STIFFNESS
The stiffness associated with the generalized displacements defined by the eigenvectors. Its value has no physical significance since the eigenvector contains an arbitrary normalizing factor but the ratio of modal stiffness to modal mass is always the eigenvalue.
MODAL TESTING
The experimental technique for measuring resonant frequencies (eigenvalues) and mode shapes (eigenvectors).

MODE PARTICIPATION FACTOR

The generalized force in each modal equation of a dynamic system.
MODE SHAPE
Same as the e igenvector. The mode shape often refers to the measure mode, found from a modal test.
MODELLING
The process of idealizing a system and its loading to produce a numerical (finite element) model.
MODIFIED NEWTON-RAPHSON
A form of the Newton-Raphson process f or solving non-linear equations where the tangent stiffness matrix is held constant for some steps. It is suitable for mildly non-linear problems.
MOHR COULOMB EQUIVALENT STRESS
A form of equivalent stress that includes the effects of friction in granular (e.g. sand) materials.
MULTI DEGREE OF FREEDOM
The system is defined by more than one force/displacement equation.
MULTI-POINT CONSTRAINTS
Where the constraint is defined by a relationship between more than one displacement at different node points.

### N

NATURAL FREQUENCY
The frequency at which a structure will vibrate in the absence of any external forcing. If a model has n degrees of freedom then it has n natural frequencies. The eigenvalues of a dynamic system are the squares of the natural frequencies.
NATURAL MODE
Same as the eigenvector.
NAVIER-STOKES EQUATIONS
Partial differential equations defining the unsteady viscous flow of fluids.
NEWMARK METHOD NEWMARK BETA METHOD
An implicit solution method for integrating second order equations of motion. It can be made unconditionally stable.
NEWTON COTES FORMULAE
A family of methods for numerically integrating a function.
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.
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.
NODE NODES NODAL
The point at which one element connects to another or the point where an element meets the model boundary. Nodes allow internal loads from one element to be transferred to another element. Element behavior 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 behavior 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.
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 SYSTEM 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.
NUMERICAL INTEGRATION
The process of finding the approximate integral of a function by numerical sampling and summing. In the finite element method the element matrices are usually formed by the Gaussian quadrature form of numerical integration.

Finite Element Analysis FEA Terms and Definitions (A to Z) Part-5

###### REFERENCES

http://www.cae.tntech.edu/~chriswilson/FEA/ANSYS/

www.nafems.org

## Finite Element Analysis FEA Terms and Definitions (A to Z) Part-3

##### (A to Z) of Finite Element Analysis

Finite Element Analysis FEA Terms and Definitions (A to Z) Part-2

### G

GAP ELEMENT CONTACT ELEMENT
These are special forms of non-linear element that have a very high stiffness in compression and a low stiffness in tension. They are used to model contact conditions between surfaces. Most of these elements also contain a model for sliding friction between the contacting surfaces. Some gap elements are just line springs between points and
others are more general forms of quadrilateral or brick element elements. The line spring elements should only be used in meshes of first order finite elements.

GAUSS POINT EXTRAPOLATION GAUSS POINT STRESSES

Stresses 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.
GAUSS POINTS GAUSS WEIGHTS
The Gauss points are the sample points used within the elements for the numerical integration of the matrices and loadings. They are also the points at which the stresses can be recovered. The Gauss weights are associated factors used in the numerical integration process. They represent the volume of influence of the Gauss points. The
positions of the Gauss points, together with the associated Gauss weights, are available in tables for integrations of polynomials of various orders.
GAUSSIAN ELIMINATION
A form of solving a large set of simultaneous equations. Within most finite element systems a form of Gaussian elimination forms the basic solution process.
A 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.
GENERALIZED COORDINATES
A set of linearly independent displacement coordinates which are consistent with the constraints and are just sufficient to describe any arbitrary configuration of the system. Generalized coordinates are usually patterns of displacements, typically the system eigenvectors.
GENERALIZED MASS
The mass associated with a generalized displacement.
GENERALIZED STIFFNESS
The stiffness associated with a generalized displacement.
GEOMETRIC PROPERTIES
Various shape dependent properties of real structures, such as thickness, cross sectional area, sectional moments of inertia, centroid location and others that are applied as properties of finite elements.
GEOMETRIC STIFFNESS STRESS STIFFNESS
The 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 ERRORS
Errors in the geometrical representation of the model. These generally arise from the approximations inherent in the finite element approximation.
GLOBAL STIFFNESS MATRIX
The assembled stiffness matrix of the complete structure.
GROSS DEFORMATIONS
Deformations sufficiently high to make it necessary to include their effect in the solution process. The problem requires a large deflection non-linear analysis.
GUARD VECTORS

The subspace iteration (simultaneous 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.
GUYAN REDUCTION METHOD
A 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 FORCES
Forces arising from Coriolis acceleration. These can destabilize a dynamic response and cause whirling.

### H

HARDENING STRUCTURE
A structure where the stiffness increases with load.
A dynamic loading that is periodic and can be represented by a Fourier series.
HEAT CONDUCTION
The analysis of the steady state heat flow within solids and fluids. The equilibrium balance between internal and external heat flows.
HERMITIAN SHAPE FUNCTIONS
Shape functions that provide both variable and variable first derivative continuity (displacement and slope continuity in structural terms) across element boundaries.
HEXAHEDRON ELEMENTS
Type of 3D element which has six quadrilateral faces.
HIDDEN LINE REMOVAL
Graphical plots of models where non-visible mesh lines are not plotted.
HIGH ASPECT RATIO LOW ASPECT RATIO
The ratio of the longest side length of a body to the shortest is termed its aspect ratio. Generally bodies with high aspect ratios (long and thin) are more ill conditioned for numerical solution than bodies with an aspect ratio of one.
HOLONOMIC CONSTRAINTS
Constraints that can be defined for any magnitude of displacement.
HOOKES LAW
The material property equations relating stress to strain for linear elasticity. They involve the material properties of Young’s modulus and Poisson ratio.
HOURGLASS MODE
Zero energy modes of low order quadrilateral and brick elements that arise from using reduced integration. These modes can propagate through the complete body.
H-CONVERGENCE
Convergence towards a more accurate solution by subdividing the elements into a number of smaller elements. This approach is referred to as H – convergence because of improved discretization due to reduced element size.
H-METHOD

A finite element method which requires an increasing number of elements to improve the solution.
H-REFINEMENT P-REFINEMENT
Making the mesh finer over parts or all of the body is termed h -refinement. Making the element order higher is termed p -refinement.

HYBRID ELEMENTS
Elements that use stress interpolation within their volume and displacement interpolation around their boundary.
HYDROSTATIC STRESS
The stress arising from a uniform pressure load on a cube of material. It is the average value of the direct stress component s at any point in the body.
HYSTERETIC DAMPING
A damping model representing internal material loss damping. The energy loss per unit cycle is independent of frequency. It is only valid for harmonic response.

### I

ILL-CONDITIONING ERRORS
Numerical (rounding) errors that arise when using ill- conditioned equations.
ILL-CONDITIONING ILL-CONDITIONED EQUATIONS
Equations 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.
IMPULSE RESPONSE FUNCTION
The response of the system to an applied impulse.
IMPULSE RESPONSE MATRIX
The 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.
INCREMENTAL SOLUTION
A solutions process that involves applying the loading in small increments and finding the equilibrium conditions at the end of each step. Such solutions a re generally used for solving non-linear problems.
INELASTIC MATERIAL BEHAVIOR
A material behavior where residual stresses or strains can remain in the body after a loading cycle, typically plasticity and creep.
INERTANCE (ACCELERANCE)
The ratio of the steady state acceleration response to the value of the forcing function for a sinusoidal excitation.
INERTIA FORCE
The force that is equal to the mass times the acceleration.
INITIAL BUCKLING
The load at which a structure first buckles.
INITIAL STRAINS

The components of the strains that are non-elastic. Typically thermal strain and plastic strain.
INTEGRATION BY PARTS
A method of integrating a function where high order derivative terms are partially integrated to reduce their order.
INTERPOLATION FUNCTIONS SHAPE FUNCTIONS
The 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 ELEMENT
Elements that use the same shape functions (interpolations) to define the geometry as were used to define the displacements. If these elements satisfy the convergence requirements of constant stress representation and strain free rigid body motions for one geometry then it will satisfy the conditions for any geometry.
ISOTROPIC MATERIAL
Materials where the material properties are independent of the co -ordinate system.

### J

JACOBI METHOD
A method for finding eigenvalues and eigenvectors of a symmetric matrix.
JACOBIAN MATRIX
A square matrix relating derivatives of a variable in one coordinate system to the derivatives of the same variable in a second coordinate system. It arises when the chain rule for differentiation is written in matrix form.
J-INTEGRAL METHODS
A method for finding the stress intensity factor for fracture mechanics problems.
JOINTS
The interconnections between components. Joints can be difficult to model in finite element terms but they can significantly affect dynamic behavior.

### K

KINEMATIC BOUNDARY CONDITIONS
The necessary displacement boundary conditions for a structural analysis. These are the essential boundary conditions in a finite element analysis.
A method for finding equivalent nodal loads when the actual load is distributed over a surface of a volume. The element shape functions are used so that the virtual work done by the equivalent loads is equal to the virtual work done by the real loads over the same virtual displacements. This gives the most accurate load representation for the finite element model. These are the non-essential stress boundary conditions in a finite element analysis.
KINEMATICALLY EQUIVALENT MASS
If the mass and stiffness are defined by the same displacement assumptions then a kinematically equivalent mass matrix is produced. This is not a diagonal (lumped) mass matrix.
KINETIC ENERGY

The energy stored in the system arising from its velocity. In some cases it can also be a function of the structural displacements.