FEM Self Supporting Course

Introduction to Finite Element Methods

Dr R MeenakumariProfessor

KEC

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Introduction• Field of electrical engg

– Theoretical electricity/electromagnetism• Laws and principles of electromagnetism

– Applied electricity/electromagnetism• Construction of mathematical models of physical

phenomena– Computational electromagnetism

• Solving specific problems by simulation through numerical methods

Ken Youssefi Mechanical Engineering Dept

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Why FEM analysis?• Complex geometry• Mixed set of materials involved. Will have

nonlinear characteistic also.• Mixed phenomena

– Electromagnetic aspects– Thermal aspects – Mechanical aspects– Dynamical aspects

• Numerical solution is the best one

Ken Youssefi Mechanical Engineering Dept

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Steps involved in analysis• Modelling

– Selection of aspects ( For eg: steady state or transient state etc.,,)

• Selection of numerical methods– Finite difference method

• Linear problems with regular geometry and time dependent system

– Boundary element method– Moment method– Montocarlo method– Finite element method

• Most widely used method both for linear and nonlinear –no restrictions on the geometry

Ken Youssefi Mechanical Engineering Dept

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Linear cases and simple geometry

Without meshing

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History of FEM?

Finite Element Analysis (FEA) was first developed in 1943 by R. Courant, who utilized the Ritz method of numerical analysis and minimization of variational calculus.

A paper published in 1956 by M. J. Turner, R. W. Clough, H. C. Martin, and L. J. Topp established a broader definition of numerical analysis. The paper centered on the "stiffness and deflection of complex structures".

By the early 70's, FEA was limited to expensive mainframe computers generally owned by the aeronautics, automotive, defense, and nuclear industries. Since the rapid decline in the cost of computers and the phenomenal increase in computing power, FEA has been developed to an incredible precision.

Ken Youssefi Mechanical Engineering Dept

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What is FEM?

• Mathematical method for solving ordinary and partial differential equation

• Able to solve complex problem that can be represented by differential equation

• A very important tool for those application which involved engineering design

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FEA Applications

•  Structural strength design• Structural interaction with fluid flows• Analysis of shock(underwater & in materials)• Acoustics• Thermal analysis• Crash simulation • Electrical analysis• Electromagnetic evaluations• Metal forming

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FEM in Electrical Engineering

•  Used to analyse linear electric and magnetic behavior

• Evaluation of electric, magnetic and thermal effects

• May be static, harmonic or transient in nature

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FEA Applications

•  Rotating machines– DC machines, Synchronous machines, induction

machines, Stepper motor, PM motors, BLDC motors, SRM motors etc.,

• Energy transfer and conversion modules– Transformers, Cables, High voltage devices,

insulators• Electrical actuators

– Linear motors, Electromagnetic brakes, magnetic bearings

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FEA Applications

•  Sensors– Capacitive and inductive , speed

• Field generators – Magnetic recording– Mass spectrometer

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Analysis Types•  Magnetostatic anlaysis

– Magnetic analysis of solenoids, electric motors, magnetic disk drives

– Magnetic flux density, field intensity, forces, torque, inductance and flux linkage

• Transient Electromagnetic analysis– Transient or steady state analysis designing for

various AC or DC devices– Time functions of magnetic flux density, field

intensity, forces, torque

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Analysis Types• Time harmonic electromagnetic analysis

– Analysis of magnetic fields caused by alternating currents

– Electric current , Voltage, generated Joule heat, impedances and inductances

• Electrostatic analysis– Fuses, transmission lines etc.,– Voltages, electric fields, capacitances

• Current flow analysis– Variety of conductive system– Voltages, current densities, power loss

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Analysis Types• Thermal analysis

– Design of many electrical and mechanical systems

– Temperature distribution, thermal gradients,heat loss

• Stress analysis– Design of many electrical and mechanical

components– Displacements, strains and stresses

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Which differential equation in EE?

2D or 3D analysis ?

Ken Youssefi Mechanical Engineering Dept

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Finite element analysis starts with an approximation of the region of interest into a number of meshes (triangular elements). Each mesh is connected to associated nodes (black dots) and thus becomes a finite element.

Basics of Finite Element Analysis

Consider a cantilever beam shown.

Symmetries for analysis

• Plane symmetry

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• Axial symmetry

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Basics of Finite Element Analysis• After approximating the object by finite elements,

each node is associated with the unknowns to be solved.

• For the cantilever beam the displacements in x and y would be the unknowns.

• This implies that every node has two degrees of freedom and the solution process has to solve 2n degrees of freedom.

• Once the displacements have been computed, the strains are derived by partial derivatives of the displacement function and then the stresses are computed from the strains.

Ken Youssefi Mechanical Engineering Dept

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Finite Element Analysis

• Pre-Processing• Solving Matrix (solver)• Post-Processing

FEA requires three steps

FEA is a mathematical representation of a physical system and the solution of that mathematical representation

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FEA Pre-ProcessingMesh

Mesh is your way of communicating geometry to the solver, the accuracy of the solution is primarily dependent on the quality of the mesh.

The better the mesh looks, the more accurate the solution is.

A good-looking mesh should have well-shaped elements, and the transition between densities should be smooth and gradual without skinny, distorted elements.

Ken Youssefi Mechanical Engineering Dept

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CAD Modeling for FEA

• CAD models prepared by the design group for eventual FEA.

• CAD models prepared without consideration of FEA needs.

• CAD models unsuitable for use in analysis due to the amount of rework required.

• Analytical geometry developed by or for analyst for sole purpose of FEA.

CAD and FEA activities should be coordinated at the early stages of the design process to minimize the duplication of effort.

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FEA Pre-ProcessingBoundary Conditions

In FEA, the name of the game is “boundary condition”, that is calculating the load and figuring out constraints that each component experiences in its working environment.

“Garbage in, garbage out”

The results of FEA should include a complete discussion of the boundary conditions.

Ken Youssefi Mechanical Engineering Dept

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Boundary Conditions

Linear Static AnalysisBoundary conditions are assumed constant from application to final deformation of system and all loads are applied gradually to their full magnitude.

Dynamic AnalysisThe boundary conditions vary with time.

Non-linear AnalysisThe orientation and distribution of the boundary conditions vary as displacement of the structure is calculated.

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Boundary Conditions

A solid face should always have at least three points in contact with the rest of the structure. A solid element should never be constrained by less than three points and only translational DOFs must be fixed.

Accuracy

The choice of boundary conditions has a direct impact on the overall accuracy of the model.

Over-constrained model – an overly stiff model due to poorly applied constraints.

Ken Youssefi Mechanical Engineering Dept

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Summary of Pre-Processing

• Build the geometry• Make the finite-element mesh• Add boundary conditions; loads and

constraints• Provide properties of material• Specify analysis type (static or dynamic,

linear or non-linear, plane stress, etc.)

These activities are called finite element modeling.

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Solving the Model - SolverOnce the mesh is complete, and the properties and boundary conditions have been applied, it is time to solve the model. In most cases, this will be the point where you can take a deep breath, push a button and relax while the computer does the work for a change.

Multiple Load and Constraint CasesIn most cases submitting a run with multiple load cases will be faster than running sequential, complete solutions for each load case.

Final Model Check

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Post-ProcessingView AnimatedDisplacements

Does the shape of deformations make sense?

View DisplacementFringe Plot

Yes

Review BoundaryConditions

No

Are magnitudes in line with your expectations?

View Stress Fringe Plot

Yes

Is the quality and mag. Of stresses acceptable?

Review Load Magnitudesand Units

No

Review Mesh Density and Quality of Elements

No

View Results SpecificTo the Analysis

Yes

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FEA - Flow Chart

Thank You

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