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Computer Aided Design (CAD)Computer Aided Design (CAD)Is based on parametric models

Copyright © 2010 J. E. Akin Rice University, MEMS Dept.

Computer Aided Design (CAD)Computer Aided Design (CAD)

Creates parametric solid models for:Creates parametric solid models for:1. Automatic multi-view drawings

ith di iwith dimensions2. Manufacturing control with

Geometric dimensioning & tolerances (GD&T)


Computer Aided Design (CAD)

Creates parametric solid models for:3 Rapid prototypes via 3D printing (wax3. Rapid prototypes via 3D printing (wax,

plastic, metals, casting sand)4 Computer Numerical Controlled (CNC)4. Computer Numerical Controlled (CNC)

machining from an initial solid block



Creates parametric solid model data for:5 Mass property calculations (surface area5. Mass property calculations (surface area,

volume, centroid, moment of inertia)6 Product Data Management (PDM)6. Product Data Management (PDM)



• Creates parametric solid model data to feed analysis systems such as automatic mesh y ygenerators, finite element analysis (FEA), finite volume analysis (FVA), finite y ( ),difference methods (FDM), mechanism motion and kinetics studies, etc.,


CAD and Finite Element Analysis

• Most ME CAD applications require a FEA in one or more areas:– Stress Analysis– Thermal AnalysisThermal Analysis– Vibrations or Structural Dynamics– Computational Fluid Dynamics (CFD)Computational Fluid Dynamics (CFD)– Electromagnetic Analysis


– ...

FEA Data Reliability• Geometry: generally most accurate• Geometry: generally most accurate• Material: accurate if standardized• Mesh: requires engineering judgement• Loads: less accurate, require assumptions, q p• Restraints: least accurate, drastically

effects results; several reasonable restrainteffects results; several reasonable restraint cases should be studied


Primary FEA Assumptions• Model geometry• Material Properties• Material Properties

– Elastic modulus, Thermal Conductivity, etc. M h( )• Mesh(s)– Element type and size, size transition rates

• Source (Load) Cases– Types of loads, Factors of safety, Coord. Sys.

• Boundary conditions (Fixtures)– Coordinate system(s)


Coordinate system(s)

General Approach for FEAGeneral Approach for FEA• Select verification tools to check the study

– analytic, experimental, other FEA method, etc.

• Understand the primary variables (PV) in the differential equation

• Understand boundary conditions (BC)– Essential, or Dirichlet BC on PV in the original

differential equation at a boundary (EBC)N t l N BC i l b d– Natural, or Neumann BC in lower space boundary differential equation (NBC)

• One or the other applied at a boundary pointCopyright © 2010 J. E. Akin Rice University, MEMS Dept.

One or the other applied at a boundary point

General Approach for FEA, 2

• Understand secondary variables (SV) obtained from the gradient of the primary g p yvariables and usually combined with the material propertiesp p– Statics: strains, stresses, failure criterion– Thermal: temperature gradient, heat fluxThermal: temperature gradient, heat flux


General Approach for FEA, 3

• Understand boundary conditions (BC)– Essential, or Dirichlet BC (on PV)Essential, or Dirichlet BC (on PV)

• Statics: displacement and/or (maybe) rotation• Thermal: temperature

– Natural, or Neumann BC (on SV)• Statics: zero surface traction vector• Thermal: zero normal heat flux

• One or the other at a boundary point.


FEA Accuracy

• PV are most accurate at the mesh nodes.• SV are least accurate at the mesh nodesSV are least accurate at the mesh nodes.

– SV are most accurate at the Gauss integration pointspoints

– SV can be post-processed for accurate nodal values (and error estimates)va ues (a d e o est ates)


General Approach for FE 1General Approach for FE, 1• Select verification tools to check the study

– analytic, experimental, other fea method, etc.

• Select element type(s) and degree– 2-D, 3-D solid, axisymmetric solid, thick surface,

thin surface, thick curve, thin curve, etc.

U d t d i i bl (PV)• Understand primary variables (PV)– Stress Analysis: displacements & (maybe) rotations

Thermal Analysis: temperature– Thermal Analysis: temperature– Fluid Flow: velocity & pressure


General Approach for FE, 4

• Understand reactions needed to maintain the Essential BC– Statics:

• Force at given displacementg p• Moment at given rotation (if active)

– Thermal:• Heat flux at given temperature


General Approach for FE, 5

• 1. Estimate the solution results and gradients• 2. Select an acceptable error (1 %)p ( )• 3. Mesh the model• 4. Load the model and apply essential BCpp y• 5. Solve the model (PV), then post-process (SV)• 6. Estimate the error levels6. Estimate the error levels

– A. Unacceptable error: Adapt mesh, go to 3– B. Acceptable error: Validate the analysis


p y

FE Mesh (FEM)• Crude meshes that “look like” a part are ok• Crude meshes that look like a part are ok

for images and mass properties but not for FE analysis.y

• Local error is proportional to product of the local mesh size (h) and the gradient of the ( ) gsecondary variables.

• PV piecewise continuous polynomials of degree p, and SV are discontinuous polynomials of degree (p-1).


FEA Stress Models• 3-D Solid, PV: 3 displacements (no rotations),

SV 6 (3 l & 3 h )SV: 6 stresses (3 normal & 3 shear stresses)• 2-D Approximations

– Plane Stress (σzz = 0) PV: 2 displacements, SV: 3 stresses

– Plane Strain (ε zz = 0) PV: 2 displacements, SV: 3 stresses (and σzz from Poisson’s ratio)

– Axisymmetric (∂/∂θ = 0) PV: 2 displacements, SV: 4 stresses


FEA Stress Models, 2,

• 2-D Approximations2 D Approximations – Thick Shells, PV: 3 displacements (no

rotations), SV: 5 (or 6) stresses), ( )– Thin Shells, PV: 3 displacements and 3

rotations, SV: 5 stresses (each at top, middle, ( pand bottom surfaces)

– Plate bending PV: normal displacement, in-plane rotation vector, SV: 3 stresses (each at top, middle, and bottom surfaces)


FEA Stress Models, 3FEA Stress Models, 3• 1-D Approximationspp

– Bars (Trusses), PV: 3 displacements (1 local axial displacement), SV: 1 axial stress

– Torsion member, PV: 3 rotations (1 local axial rotation), SV: 1 torsional stress B (F ) PV 3 di l t 3– Beams (Frames), PV: 3 displacements, 3 rotations, SV: axial, bending, & shear stress

• Thick beam, thin beam, curved beamThick beam, thin beam, curved beam• Pipe element, pipe elbow, pipe tee


Local ErrorLocal Error• The error at a (non-singular) point is the

d f h l i h hproduct of the element size, h, the gradient of the secondary variables, and

d d h d ia constant dependent on the domain shape and boundary conditions.– Large gradient points need small h– Small gradient points can have large h

• Plan local mesh size with engineering judgement of estimated gradients.


Error Estimators

• Global and element error estimates are often available from mathematical norms of the secondary variables. The energy norm is the most common.

• It is proven to be asymptotically exact for elliptical problems.elliptical problems.

• Typically we want less than 1 % error.


Error Estimates

• Quite good for elliptic problems (thermal, elasticity, ideal flow), Navier-Stokes, etc.y, ), ,

• Can predict the new mesh size needed to reach the required accuracyreach the required accuracy.

• Can predict needed polynomial degree.R i d i f• Require second post-processing pass for localized (element level) gradient smoothing.


Primary FEA Matrix Costsy• Assume sparse banded linear algebra system

of E equations with a half-bandwidth of Bof E equations, with a half-bandwidth of B.Full system if B = E.

Storage required S = B * E (Mb)– Storage required, S = B * E (Mb)– Solution Cost, C α B * E2 (time)

H lf t B B/2 E E/2 S S/4– Half symmetry: B ← B/2, E ← E/2, S ← S/4, C ← C/8 Quarter symmetry: B ← B/4 E ← E/4 S ← S/16– Quarter symmetry: B ← B/4, E ← E/4, S ← S/16, C ← C/64Eighth symmetry Cyclic symmetry


– Eighth symmetry, Cyclic symmetry, ...

Symmetry and Anti-symmetryy y y y

• Use symmetry states for the maximumUse symmetry states for the maximum accuracy at the least cost in stress and thermal problems.t e a p ob e s.

• Cut the object with symmetry planes (or surfaces) and apply new boundary(or surfaces) and apply new boundary conditions (EBC or NBC) to account for the removed materialfor the removed material.


Symmetry (Anti-symmetry)

• Requires symmetry of the geometry and material properties.p p

• Requires symmetry (anti-symmetry) of the source termsthe source terms.

• Requires symmetry (anti-symmetry) of the essential boundary conditionsthe essential boundary conditions.


Structural Model

• Symmetry– Zero displacement normal to surfaceZero displacement normal to surface– Zero rotation vector tangent to surface

• Anti-symmetry• Anti-symmetry– Zero displacement vector tangent to surface

Z t ti l t f– Zero rotation normal to surface


Thermal Model

• Symmetry– Zero gradient normal to surface (insulatedZero gradient normal to surface (insulated

surface, zero heat flux)• Anti-symmetryAnti symmetry

– Average temperature on surface known


Local SingularitiesLocal Singularities

• All elliptical problems have local radialAll elliptical problems have local radial gradient singularities near re-entrant corners in the domaincorners in the domain.

Radius, ru = r p f(θ)

∂u/∂r = r (p-1) f(θ)

Strength, p = π/C Corner: p = 2/3, weak

Crack: p = 1/2 strongRe-entrant, C

Crack: p 1/2, strong

∂u/∂r ⇒∞ as r ⇒ 0


Stress Analysis Verification, 1Stress Analysis Verification, 1

• Prepare initial estimates of deflections, reactions and stresses.

• Eyeball check the deflected shape and the principal stress vectors.p p

• Eyeball check the contour lines for wiggles (OK in low stress regions )wiggles. (OK in low stress regions.)


St A l i V ifi ti 2Stress Analysis Verification, 2

• The stresses often depend only on the shape of the part and are independent of the material properties.

• You must also verify the displacements y pwhich almost always depend on the material properties.p p


Stress Analysis Verification, 3

• The reaction resultant forces and/or moments are equal and opposite to the q ppactual applied loading.

• For pressures or tractions remember toFor pressures or tractions remember to compare their integral (resultant) to the solution reactions.solution reactions.

• Reactions can be obtained at elements too.


Stress Analysis Verification, 4

• Compare displacements, reactions and stresses to initial estimates. Investigate any differences.

• Check maximum error estimates, if ,available in the code.


Thermal Analysis Verification 1Thermal Analysis Verification, 1

• Prepare initial estimates of thePrepare initial estimates of the temperatures, reaction flux, and heat flux vectorsflux vectors.

• Eyeball check the temperature contours and the heat flux vectorsand the heat flux vectors.

• Temperature contours should be di l i l d b dperpendicular to an insulated boundary.


Thermal Analysis Verification, 2

• The temperatures often depend only on the shape of the part.

• Verify the heat flux magnitudes which almost always depend on the material y pproperties.


Thermal Analysis Verification 3Thermal Analysis Verification, 3

• The reaction resultant nodal heat fluxes are equal and opposite to the applied heat fluxes.q pp pp

• For distributed heat fluxes remember to compare their integral (resultant) to thecompare their integral (resultant) to the solution reactions.

• Reactions can be obtained at elements too• Reactions can be obtained at elements too.


Thermal Analysis Verification, 4

• Compare temperatures, reactions and heat flux vectors to initial estimates. Investigate any differences.

• Check maximum error estimates, if ,available in the code.


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