Finite Element Analysis

Praveen Patil

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Contents

• Introduction to the Finite Element Method

(FEM)

••• Future Future Future TrendsTrendsTrends

FEM Applied to Solid Mechanics Problems

Create elementsof the beam

dxi 1 dxi 2

dyi 1 dyi 21 2

4 3

Nodal displacement and forces

• A FEM model in solid mechanicscan be thought of as a system ofassembled springs. When a loadis applied, all elements deformuntil all forces balance.

• F = Kd

• K is dependant upon Young’s modulus and Poisson’s ratio, as well as the geometry.

• Equations from discrete elements are assembled together to form the global stiffness matrix.

• Deflections are obtained by solving the assembled set of linear equations.

• Stresses and strains are calculated from the deflections.

Classification of Solid-Mechanics Problems

Analysis of solids

Static Dynamics

Behavior of Solids

Linear Nonlinear

Material

Fracture

GeometricLarge 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

Governing Equation for Solid Mechanics Problems

[K] {u} = {Fapp} + {Fth} + {Fpr} + {Fma} + {Fpl} + {Fcr} + {Fsw}

+ {Fld}

[K] = total stiffness matrix

{u} = nodal displacement

{Fapp} = applied nodal force load vector

{Fth} = applied element thermal load vector

{Fpr} = applied element pressure load vector

{Fma} = applied element body force vector

{Fpl} = element plastic strain load vector

{Fcr} = element creep strain load vector

{Fsw} = element swelling strain load vector

{Fld} = element large deflection load vector

• Basic equation for a static analysis is as follows:

Six Steps in the Finite Element Method

• Step 1 - Discretization: The problem domain is discretized into a collection of simple shapes, or elements.

• Step 2 - Develop Element Equations: Developed using the physics of the problem, and typically Galerkin’s Method or variational principles.

• Step 3 - Assembly: The element equations for each element in the FEM mesh are assembled into a set of global equations that model the properties of the entire system.

• Step 4 - Application of Boundary Conditions: Solution cannot be obtained unless boundary conditions are applied. They reflect the known values for certain primary unknowns. Imposing the boundary conditions modifies the global equations.

• Step 5 - Solve for Primary Unknowns: The modified global equations are solved for the primary unknowns at the nodes.

• Step 6 - Calculate Derived Variables: Calculated using the nodal values of the primary variables.

Process Flow in a Typical FEM Analysis

StartProblemDefinition

Pre-processor

• Reads or generates nodes and elements (e.g. MD-Patran)

• Reads or generates material property data.

• Reads or generates boundary conditions (loads and constraints.)

Processor/Solver

• Generates element shape functions

• Calculates master element equations

• Calculates transformation matrices

• Maps element equations into global system

• Assembles element equations

• Introduces boundary conditions

• Performs solution procedures

Post-processor

• Prints or plots contours of stress components.

• Prints or plots contours of displacements.

• Evaluates and prints error bounds.

Analysis anddesign decisions

Stop

Step 1, Step 4

Step 6

Steps 2, 3, 5

Step 1: Discretization - Mesh Generation

airfoil geometry(from CAD program e.g CATIA)

e.g. MD-Patran

surface model

ET,1,SOLID45N, 1, 183.894081 , -.770218637 , 5.30522740N, 2, 183.893935 , -.838009645 , 5.29452965..TYPE, 1E, 1, 2, 80, 79, 4, 5, 83, 82E, 2, 3, 81, 80, 5, 6, 84, 83...

meshed model

Step 4: Boundary Conditions for a Solid Mechanics Problem

• Displacements ⇒ DOF constraints usually

specified at model boundaries to define rigid

supports.

• Forces and Moments ⇒ Concentrated loads on

nodes usually specified on the model exterior.

• Pressures ⇒ Surface loads usually specified on

the model exterior.

• Temperatures ⇒ Input at nodes to study the

effect of thermal expansion or contraction.

• Inertia Loads ⇒ Loads that affect the entire

structure (ex: acceleration, rotation).

Step 4: Applying Boundary Conditions (Thermal Loads)

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150175

Tempmapper

Nodes fromFE Modeler

Thermal Soln Files

bf, 1,temp, 149.77bf, 2,temp, 149.78...bf, 1637,temp, 303.64bf, 1638,temp, 303.63

Step 4: Applying Boundary Conditions (Other Loads)

• Speed, temperature and hub fixity applied to sample

problem.

• FE Modeler used to apply speed and hub constraint.

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antype,staticomega,10400*3.1416/30d,1,all,0,0,57,1

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 analysis

»Temperature

» Heat fluxes

» Thermal gradients

» Heat flow from convection faces

• Fluid analysis

» Pressures

» Gas temperatures

» Convection coefficients

» Velocities

Example FEM Application Areas

• Automotive 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 scientific and technological problems.

• FEM is increasingly applied to a variety of real-world design and analysis problems.

Technologies That Compete With the FEM

• Other numerical solution methods:

– Finite differences

» Approximates the derivatives in the differential equation using

difference equations.

» Useful for solving heat transfer and fluid mechanics problems.

» Works well for two-dimensional regions with boundaries parallel

to the coordinate axes.

» Cumbersome when regions have curved boundaries.

– Weighted residual methods (not confined to a small subdomain):

» Collocation

» Subdomain

» Least squares*

» Galerkin’s method*

– Variational Methods* (not confined to a small subdomain)

* Denotes a method that has been used to formulate finite element

solutions.

Technologies that Compete With the FEM (cont.)

• Prototype Testing

» Reliable. Well-understood.

» Trusted by regulatory agencies (FAA, DOT, etc.)

» Results are essential for calibration of simulation software.

» Results are essential to verify modeled results from simulation.

» Non destructive testing (NDT) is lowering costs of testing in

general.

» Expensive, compared to simulation.

» Time consuming.

» Development programs that rely too much on testing are

increasingly less competitive in today’s market.

» Faster product development schedules are pressuring the quality of

development test efforts.

» Data integrity is more difficult to maintain, compared to

simulation.

Contents

••• Introduction to the Finite Element Method Introduction to the Finite Element Method Introduction to the Finite Element Method

(FEM)(FEM)(FEM)

• Future Trends

Future Trends in the FEM and Simulation

• The FEM in particular, and simulation in general, are becoming

integrated with the entire product development process (rather than just

another task in the product development process):

– FEM cannot become the bottleneck.

• A broader range of people are using the FEM:

– Not just hard-core analysts. Future (?? Word excel??)

• 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.(??)”

Achieved overnightBIP optimization on SGI 2800/256, with

equivalent yield of 9 months CPU time

NVH & Crash Optimization of Vehicle Body Overnight

• Ford body-in-prime (BIP) model of 390K DOF

• MSC.Nastran for NVH, 30 design variables

• RADIOSS for crash, 20 design variables

• 10 design variables in common

• Sensitivity based Taylor approx. for NVH

• Polynomial response surface for crash

Conflicting Variables . . .with Reducing time

Future Trends in the FEM and Simulation (cont.)

• Enhanced multiphysics capabilities are coming:

– Coupling between numerous physical phenomena.

» Ex: Fluid-structural interaction is the most common example.

» Ex: Semiconductor circuits, EMI and thermal buildup vary with current

densities.

• Improved life predictors, improved service estimations.

• 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.

• FEM and simulation software available via Internet subscription.

• Decreasing reliance on testing. But (??)

Workstationsand Servers

Mainframes

Economics: Physical prototyping costs continue IncreasingEngineer more expensive than simulation tools

1960 2006Years

Cost of CAESimulation

Cost of PhysicalPrototyping

Cost of CAEEngineer

MSC/NASTRANSimulation Costs

(Source: General Motors)

MSC/NASTRANSimulation Costs

(Source: General Motors)

CAE Engineervs. System Costs(Source: Detroit Big3)

CAE Engineervs. System Costs(Source: Detroit Big3)

1960$30,000

1960$30,000

Engineer$36/hr

Engineer$36/hr

2006$0.022006$0.02

System$1.5/hrSystem$1.5/hr

Thanks.