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8/6/2019 Prof.sudi July27 APSS2010
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Fundamentals of FiniteElement Analysis
Di SuResearch assistant professorBridge & Structure Laboratory
Department of Civil EngineeringThe University of Tokyo
2010 Asia 2010 Asia - - Pacific Summer School in Smart Structures Technology Pacific Summer School in Smart Structures Technology July 27, 2010 July 27, 2010
Outline
2
Discussion of Some Key Problems
Application of FEA
Bar Element and Beam Element
FEA Concept
Introduction and History
Introduction and History
About this short courseFundamentals of finite element analysisNo a textbook of FEA, no tensor, no Galerkin methodOnly focus in Civil EngineeringRealized by Matlab, Abaqus and Ansys
Try to study FEA byMathematical principle + Analysis modeling + Software application
Try to use FEA bySoftware + Practical problem + Self development
3
FEA & Structure
4
Finite ElementFinite ElementMethodMethodDemolish
Maintenance
Construction
Design
Structure
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FEA & Structure
Beijing National Stadium (40,000 tons)
5
Beijing National Stadium
Mode shape
6From Herzog and de Meuron, Arup, CAG .
Beijing National Stadium
Failure verification
7From Herzog and de Meuron, Arup, CAG .
Beijing National Stadium
Truss column design
8From Herzog and de Meuron, Arup, CAG .
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Beijing National Stadium
Construction process
9From Herzog and de Meuron, Arup, CAG .
Beijing National Stadium
Construction process
10From Herzog and de Meuron, Arup, CAG .
Finite Element Method Defined
Complexities in the geometry, properties and in the boundaryconditions that are seen in most real-world problems usuallymeans that an exact solution cannot be obtained or obtained ina reasonable amount of time.
Engineers are content to obtain approximate solutions that canbe readily obtained in a reasonable time frame, and withreasonable effort. The FEM is one such approximate solutiontechnique.The FEM is a numerical procedure for obtaining approximatesolutions to many of the problems encountered in engineeringanalysis.
11
Discretized approximation
12
Rayleigh-Ritz principle Approximation in the wholedomain Higher-order continuousfunction Fewer base functions
Another method Pieces functionapproximation in sub-domain Linear or polynomial function More base functions
Basic idea of FEM
Describe one complexfunction
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Finite Element Method Definition
The continuum has an infinite number of degrees-of-freedom(DOF), while the discretized model has a finite number of DOF. This is the origin of the name, finite element method.
The number of equations is usually rather large for most real-world applications of the FEM, and requires the computationalpower of the digital computer. The FEM has little practicalvalue if the digital computer were not available .
Solution of FEM gives the approximate behavior of thecontinuum or system.
13
The concept of FINITE
14
Finite NumberThere is only finitenumber of elementsin your analysis
model, not infinite.
FINITE
Finite AccuracyThe accuracy of your
analysis is finite. Evenfor very fine model, it
is not accuratesolution.
History of FEM
15
Engineering Mathematics
Trial function Finite differencemethodVariational
methodSimilar structure
replacement
Method of Weighted
Residuals
Continuous trialfunction
Direct continuumelements
Variable finitedifference method
Present FiniteElement Method
Rayleigh 1870Ritz 1909 Gauss 1795Galerkin 1915
Biezeno-Koch 1923
Richardson 1910Liebman 1918Southwell 1946
Hrenikoff 1941Mchenry 1943Newmark 1949
Courant 1943Prager-Synge 1947Zienkiewicz 1964
Argyris 1955Turner et al. 1956
Varga 1962
First coined by Clough 1960
History of FEM
It is difficult to document the exact origin of the FEM, because the basicconcepts have evolved over a period of 150 or more years. The first book on the FEM by Zienkiewicz and Chung was published in 1967.
Most commercial FEM software packages originated in the 1970s and1980s.
The FEM is one of the most important developments in computationalmethods to occur in the 20th century. Advances in and ready availability of computers and software has brought the FEM within reach of engineersworking in small industries, and even students.
16
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FEA Concept
Example1: One dimension problem
17
Try to solve this problem?
FEA Concept
18
FEA Concept
Use u A , u B , u C as unknowns
19
FEA Concept
20
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FEA Concept
21
Load matrix Inner forcematrix
Nodaldisplacement
The equilibrium equation for whole structure, not for each component
Lets derive more
FEA Concept
22
More general form
The equilibrium for each node has turned into the relationship of each component. This component description is generalized and standard; i.e. ELEMENT . In this example, it is Bar Element.
General description of 1D bar element
23
Nodal displacement
External force
Inner force
Equilibrium equation
Stiffness matrix
Application of bar element
Could you solve this three-link structure problem usingthe bar element you just learned?
24
P3=50N
Example2:
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FEM Solution process
25
Element 1 Element 2 Element 3
Assembly
Stiffness matrix
Nodal force
Boundary condition
FEM Solution process
Solve the linear equations
Derived other parameters
26
Very standard, very simple solution, right?
Analysis modeling process
1D model 272D model 3D model
FEM Solution
28
Step 1: Discretization Step 2: Stiffness matrix foreach element
Step 3: Assembly Step 4: Solution (nodal disp.)Step5: Other parameters (strain,
stress, et al. )
Simple element
Complex structure
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Element type in FEM software
Abaqus
Ansys
29
Bar Element and Beam Element
Lets discuss the process more generally.
It will be very difficult to derive the stiffness matrix ofelement by the mechanical equations in most cases. Inthis section two general methods will be introduced toobtain the basic equation for bar element and beamelement.
Principle of virtual work
Principle of minimum potential energy
30
Bar Element
The basic parameters in x axisDisplacement: u( x)Strain: x( x)Stress: x( x)
31
Example 3: 1D problem
Bar Element
Basic equation of 1D problemEquilibrium equation or (c1 is constant)
Geometric equation
Physical equation
Boundary condition
32
How to solve?1.Direct solution: 3 unknownsfor 3 equations2.Indirect solution: Trialfunction?
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Principle of virtual work
For this equilibrium system
If a small disturbance happens,but still remains equilibrium
Principle of virtual work based on the virtual displacement
When a rigid body that is in equilibrium is subject to virtual compatible displacements, the total virtual work of all external forces is zero.
33
Virtual displacement
--Johann (Jean) Bernoulli (1667-1748) and Daniel Bernoulli (1700-1782)
Principle of virtual work
Principle of virtual work for a deformable body
If U is virtual strain energy, and W is the virtual work byexternal force
External virtual work is equal to internal virtual strain energy when equilibrated forces and stresses undergo
unrelated but consistent displacements and strains .
34
Application of principle of virtual work
Assume the displacement field as
(Trial function, c is unknown)The strain, virtual displacement, and virtual strain is
The virtual work and virtual strain energy
35
Application of principle of virtual work
From the principle of virtual work
Final solution
36
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Principle of minimum potential energy
It asserts that a structure or body shall deform or displace to a position that minimizes the total potential energy.Assume the displacement field as u( x)
Potential energy(U is the strain energy, W is the external work)
For bar element
The true displacement field should satisfy
37
Application of principle of minimum potential energy
Again,
Potential energy
From the minimum value
38
Bar element
Description of one elementGeometrics and node descriptionDisplacement field (Trial function)Strain field
Stress fieldPotential energy
Obtain the stiffness equation of element by principle ofvirtual work or principle of minimum potential energy
39
Bar element in local coordinate system
Geometrics and node description
Nodal displacementNodal forceDisplacement field
Assume the linear functionFrom the nodal displacement
Then
40
Shape function
matrix
Nodaldisplacement
vector
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Bar element in local coordinate system
Strain field
Stress field
41
Strain-displacementmatrix
Stress-displacementmatrix
Bar element in local coordinate system
Potential energy
42
Bar element in local coordinate system
Stiffness equation of bar element
43
Stiffness matrix of element
Nodal force vector
Bar element in global coordinate system
Local coordinate system
Global coordinate system
44
Transformation matrix
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Bar element in global coordinate system
Potential energy
Stiffness equation for global coordinate system
45
Bar element in space
Transformation matrix
Stiffness equation for bar element in space
46
Bar Element in MATLAB
MATLAB program for 1D bar element
Bar1D2Node _Stiffness(E,A,L)Calculate the stiffness matrix k(2 2)
Bar1D2Node _Assembly(KK,k,i,j)Assemble the stiffness matrix
Bar1D2Node _Stress(k,u,A)Calculate the stress of element
Bar1D2Node_Force(k,u)Calculate the nodal force vector
47
All the codes can be downloaded inhttp://www.bridge.t.u-tokyo.ac.jp/apss/downloads/FEM%20code.zip
Bar Element in MATLAB
MATLAB program for 2D bar element
Bar2D2Node _Stiffness(E,A,x1,y1,x2,y2,alpha)Calculate the stiffness matrix k(4 4)
Bar2D2Node _Assembly(KK,k,i,j)Assemble the stiffness matrix
Bar2D2Node _Stress(E,x1,y1,x2,y2,alpha,u)Calculate the stress of element
Bar2D2Node_Force(E,A,x1,y1,x2,y2,alpha,u)Calculate the nodal force vector
48
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Application of bar element
Example 4: Four-bar truss structure
49
Application of bar element
Stiffness matrix for each element
50
Application of bar element
Assemble to whole stiffness equation
Boundary conditions
51
Bar Element and Beam Element
Results
52
Compare with the results from MATLAB, ANSYS and ABAQUS
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Basic equation of beam element
Equilibrium equation
Geometric equation
Physical equation
57
Basic equation of beam element
Choose deflection v as the fundamental unknown
Boundary conditions
58
Equilibrium in y
Equilibrium in x
Physical
Geometric
Beam element
Description of one elementGeometrics and node descriptionDisplacement field (Trial function)Strain field
Stress fieldPotential energy
Obtain the stiffness equation of element by principle ofvirtual work or principle of minimum potential energy
59
Beam element in plane in local coordinate system
Geometrics and node description
Nodal displacementNodal force
Displacement fieldAssume the polynomial functionFrom the nodal displacement
60Shape function matrix
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Beam element in plane in local coordinate system
Strain field
Stress field
61
Strain-displacement matrix
Stress-displacement matrix
Beam element in plane in local coordinate system
Strain energy
External work
Stiffness equation 62
Stiffness matrix of element
Nodal force vector
General beam element in local coordinate system
63
Bending beam + axial deformation
Nodal displacement
Nodal force
Stiffness equation of beam element
Equivalent nodal force
How to obtain the nodal force?
64
Uniform loadDifferent BC
Equivalent nodal force
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Equivalent nodal force
Displacement field
External work
Equivalent nodal force
65
Shape function
No relationship with BC, it is a universal expression for uniform load.
Equivalent nodal force
66
Application of beam element
Example 6: Cantilever-continuous beam
67
How to obtain structural responses?
Application of beam element
Modeling using 2 beam elements
68
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Application of beam element
69No need to solve the differential equations or partialdifferential equations, just linear equations
Plane beam element in global coordinate system
Need coordinate transferLocalGlobal
70
Beam element in space
Local coordinate system
For u1 and u2 , the same with bar element
For x1 and x2, similar with bar element
For v1, v2, z1 and z2, the same with pure bending beam
For w1, w2, y1 and y2, similar as above equation71
Beam element in space
Stiffness matrix for beam element in space (localcoordinate system)
72
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Coordinate transfer in space
Transfer to global coordinate system
73
Beam Element in MATLAB
MATLAB program for 1D beam element
Beam1D2Node_Stiffness(E,I,L)Calculate the stiffness matrix k(4 4)
Beam1D2Node _Assembly(KK,k,i,j)Assemble the stiffness matrix
Beam1D2Node_ Strain(x,L,y)Calculate the geometric matrix B(1 4)
Beam1D2Node _Stress(E,B,u)Calculate the stress of element
Beam1D2Node_Deflection(x,L,u)Calculate the deflection of element
74
Beam Element in MATLAB
MATLAB program for 2D beam element
Beam2D2Node_Stiffness(E,I,A,L)Calculate the stiffness matrix k(6 6)
Beam2D2Node_Assemble(KK,k,i,j)Assemble the stiffness matrix
Beam2D2Node_Forces(k,u)Calculate the nodal force of element
75
Application of beam element
Example 7: One frame structure
76
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Application of beam element
Modeling using 3 beam elements
77
Application of beam element
For element 1, stiffness matrix is
For element 2 and 3,
78
Application of beam element
Transfer matrix for element 2 and 3
Stiffness matrix for element 2 and 3 in global coordinatesystem
Assemble the whole stiffness matrix
79
Application of beam element
After considering the BC,
Final solution
80
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MATLAB Program
81
ANSYS Program
82
Discretization for continuum elements
83
The real power of Finite Element method is that it successfullysolved the continuum problem.
Application of FEM
General-purpose FEM software packages are available atreasonable cost, and can be readily executed onmicrocomputers, including workstations and PCs.
The FEM can be coupled to CAD programs to facilitate solidmodeling and mesh generation.
Many FEM software packages feature GUI interfaces, auto-meshers, and sophisticated postprocessors and graphics tospeed the analysis and make pre and post-processing moreuser-friendly.
84
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Commercially available general FEM software
85
Year Software Company Website1965 ASKA (PERMAS) IKOSS GmbH, (INTES),Germany www.intes.de
STRUDL MCAUTO, USA www.gtstrudl.gatech.edu1966 NASTRAN MacNeal-Schwendler Corp., USA www.macsch.com
1967 BERSAFE CEGB, UK (restr uctured in 1990)SAMCEF Univer. of Liege, Belgium www.samcef.com
1969 ASAS Atkins Res.&Devel., UK www.wsasoft.com
MARC MARC Anal. Corp., USA www.marc.com
PAFEC PAFEC Ltd, UK now SER Systems
SESAM DNV, Norway www.dnv.no1970 ANSYS Swanson Anal. Syst., USA www.ansys. com
SAP NISEE, Univ. of California, Berkeley, USA www.eerc.berkeley.edu
1971 STARDYNE Mech. Res. Inc., USA www.reiusa.comTITUS (SYSTUS) CITRA, France; ESI Group www.systus.com
1972 DIANA TNO, The Netherla nds www.diana.n l
WECAN Westinghouse R&D, USA1973 GIFTS CASA/GIFTS Inc., USA
1975 ADINA ADINA R&D, Inc., USA www.adina.c omCASTEM CEA, France www.castem.org:8001/ HomePage.html
FEAP NISEE, Univ. of California, Berkeley, USA www.eerc.berkeley.edu1976 NISA Eng. Mech. Res. Corp., USA www.emrc .com1978 DYNA2D, DYNA3D Livermore Softw. Tech. Corp., USA www.lstc.com
1979 ABAQUS Hibbit, Karlsson & Sorensen, Inc., USA www.abaqus.c om1980 LUSAS FEA Ltd., UK www.lusas. com1982 COSMOS/M Structural Res. & Anal. Corp., USA www.cosmosm.com
1984 ALGOR Algor Inc., USA www.algor. com
Information Available from Various Types of FEM Analysis
Static analysis Deflection Stresses Strains Forces Energies
Dynamic analysis Frequencies Deflection (mode
shape) Stresses Strains Forces Energies
Heat transfer analysisTemperature
Heat fluxes
Thermal gradients
Heat flow fromconvection faces
Fluid analysis
Pressures
Gas temperatures
Convection coefficients Velocities
Example FEM Application AreasAutomotive industry
Static analyses Modal analyses Transient dynamics Heat transfer
Mechanisms Fracture mechanics Metal forming Crashworthiness
Aerospace industry
Static analyses
Modal analyses
Aerodynamics
Transient dynamics Heat transfer
Fracture mechanics
Creep and plasticity analyses
Composite materials Aeroelasticity
Metal forming
Crashworthiness
Architectural
Soil mechanics
Rock mechanics
Hydraulics
Fracture mechanics
Hydroelasticity
Variety of FEM Solutions is Wide and Growing Wider
The FEM has been applied to a richly diverse array of scientificand technological problems.
The next few slides present some examples of the FEM appliedto a variety of real-world design and analysis problems.
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89 90
Several examples
91From Mr. M., Chingthaka and Dr. Pellegrino, S. @Caltech
Joint expansion of aerospace structure
Sever examples
92From Jaesung Eom. et al @ Rensselaer Polytechnic Institute
Lung cancer analysis
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Several examples
Balloon inflation
From Mr. XW. Deng @Caltech
Several examples
Heat transfer analysis
94From Mr. Hida@ the University of Tokyo
Several example
Electromagnetic analysis
95From Mr. Mizutani@ the University of Tokyo
Classification of Solid-Mechanics Problems
96
Analysis of solids
Static Dynamics
Behavior of Solids
Linear Nonlinear
Material
FractureGeometric
Large Displacement
Instability
Plasticity
ViscoplasticityGeometric
Classification of solids
Skeletal Systems1D Elements
Plates and Shells2D Elements
Solid Blocks3D Elements
TrussesCablesPipes
Plane StressPlane StrainAxisymmetricPlate BendingShells with flat elementsShells with curved elements
Brick ElementsTetrahedral ElementsGeneral Elements
Elementary Advanced
Stress Stiffening
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Application of FEM
97
Example 8: Elastic-plastic analysis
Application of FEM
Example 9: Multibody system
98
99
How can the FEM Help the Design Engineer?
The FEM offers many important advantages to the design engineer :
Easily applied to complex, irregular-shaped objects composed of several different materials and having complex boundary conditions.
Applicable to steady-state, time dependent and eigenvalueproblems.
Applicable to linear and nonlinear problems.
One method can solve a wide variety of problems, includingproblems in solid mechanics, fluid mechanics, chemical reactions,electromagnetics, biomechanics, heat transfer and acoustics, to namea few.
100
How can the FEM Help the Design Organization?
Simulation using the FEM also offers important business advantages tothe design organization :
Reduced testing and redesign costs thereby shortening the productdevelopment time.
Identify issues in designs before tooling is committed.
Refine components before dependencies to other componentsprohibit changes.
Optimize performance before prototyping.
Discover design problems before litigation.
Allow more time for designers to use engineering judgment, and less
time turning the crank.
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Disadvantages of the Finite Element Method
Numerical problems: Computers only carry a finite number of significant digits. Round off and error accumulation. Can help the situation by not attaching stiff (small) elements
to flexible (large) elements.Susceptible to user-introduced modeling errors:
Poor choice of element types. Distorted elements. Geometry not adequately modeled.
Certain effects not automatically included: Buckling Large deflections and rotations. Material nonlinearities . Other nonlinearities.
105
Sources of Error in the FEM
The three main sources of error in a typical FEM solution arediscretization errors, formulation errors and numerical errors.
Discretization error results from transforming the physical system(continuum) into a finite element model, and can be related tomodeling the boundary shape, the boundary conditions, etc.
106
Sources of Error in the FEM
Formulation error results from the use of elements that don't precisely describe thebehavior of the physical problem.Elements which are used to model physical problems for which they are not suited aresometimes referred to as ill-conditioned or mathematically unsuitable elements.For example a particular finite element might be formulated on the assumption thatdisplacements vary in a linear manner over the domain. Such an element will produceno formulation error when it is used to model a linearly varying physical problem (linearvarying displacement field in this example), but would create a significant formulationerror if it used to represent a quadratic or cubic varying displacement field.
107
Sources of Error in the FEM
Numerical error occurs as a result of numericalcalculation procedures, and includes truncation errors andround off errors.
Numerical error is therefore a problem mainly concerningthe FEM vendors and developers.
The user can also contribute to the numerical accuracy,for example, by specifying a physical quantity, sayYoungs modulus, E, to an inadequate number of decimalplaces.
108
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Stiffening and lower bound
The finite element method (FEM) provides a lower boundin energy norm for the exact solution, i.e., theapproximation solution (displacement field) from FEM issmaller than actual case.
This is simply explained like this. FEM uses a finitenumber of DOF to describe the continuum which has aninfinite number of DOF. This will made the stiffness ofsystem increase (stiffening), therefore, displacement willbecome small for the same external force.
109
High-order element
Using a different set of shape functions of high-order polynomials willexpect to reduce the computational effort and increase the accuracyof the results. It can provide that an increase of polynomial degree iscombined with a proper mesh design.
110
2 nodes, linear function
3 nodes, quadratic function 4 nodes, cubic function
h-method vs p-method
h-methodThe basis functions for each finite element can be refined and thediameter of the largest element, hmax , allowed to approach zero. Thismode is called h-convergence and its computer implementation theh-version or h-method of the finite element method.
111
Defined in Ivo Babuska, Barna Szabo, On the rates of convergence of the finite element method, International Journal for Numerical Methods in Engineering, 18(3):323-341, 2005 .
h-method vs p-method
p-methodThe finite element mesh can be refined and the minimal order of (polynomial) basis functions, pmin , allowed to approach infinity.This mode is called p-convergence and its computer implementationthe p-version or p-method of the finite element method.
112
Defined in Ivo Babuska, Barna Szabo, On the rates of convergence of the finite element method, International Journal for Numerical Methods in Engineering, 18(3):323-341, 2005 .
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h-method vs p-method
Which method is better? No conclusionIn the p-version of the finite element method the rate of convergence cannot be slower than in the h-version.Numerical oscillation problem would happen for p-version of thefinite element method.For obvious practical reasons, finite element analyses should beboth efficient and reliable.
My personal viewFor structural analysis, h-method is more popular. Two-orderelement is a good application considering the efficiency andreliability.p-method seems to act against the original goal of FEM.
113
h-method vs p-method
Really? From p-version FEM software Stresscheck
114
Up to 8-order element???
From http://www.ada.co.jp/products/StressCheck/sc_pfem.html
General software vs specific software
My personal view:It is very important to make the FEM program by oneselfwhen studying the FEM.
For normal use of FEA, general software is morerecommendable. The current software has been well-developed and ready to handle all the problems
Even for very specific problems, plenty of user-definedsubroutines can be used.
Comparing with maintenance of one whole analysisprogram, just to maintain one specific part of the programwill be more focused and efficient. 115
General software vs specific software
116
User Subroutine in ABAQUS
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The FEM in particular, and simulation in general, are becomingintegrated with the entire product development process (rather than
just another task in the product development process).
A broader range of people are using the FEM.
Increased data sharing between analysis data sources (CAD, testing,FEM software, ERM software.)
FEM software is becoming easier to use:Improved GUIs, automeshers.Increased use of sophisticated shellscripts and wizards.
117
Future Trends in the FEM and Simulation
Enhanced multiphysics capabilities are coming:Coupling between numerous physical phenomena.
Ex: Fluid-structural interaction is the most common example.
Increasing use of non-deterministic analysis and design methods:Statistical modeling of material properties, tolerances, and anticipated loads.Sensitivity analyses.
Faster and more powerful computer hardware. Massively parallel processing. Ex: ADVENTURE PROJECT @ the University of Tokyo.
Decreasing reliance on testing.
FEM and simulation software available freely. Ex: OpenSees @ University of California, Berkeley .
Ex: ADVENTURE PROJECT @ the University of Tokyo.118
Future Trends in the FEM and Simulation
Suggested reference
Chandrupatla, T. R. and Ashok D. Belegundu, 1997. Introduction to Finite Elementsin Engineering , Prentice Hall, Upper Saddle River, New Jersey.Kardestuncer, H., 1987. Finite Element Handbook , McGraw-Hill, New York.Segerlind, L. J., 1984. Applied Finite Element Analysis , John Wiley and Sons, New
York.Chandrupatla, Tirupathi R., 2002 . Introduction to finite elements in engineering ,Prentice Hall, Third Edition.R2. O. C. Zienkiewicz, R. L. Taylor and J. Z. Zhu, 2005. The Finite Element Method:
Its Basis and Fundamentals , Elsevier Butterworth Heinemann, Sixth Edition.Pan Zeng, 2008. Fundamentals of Finite Element Analysis , Tsinghua University.
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