Is based on parametric models. J. E. Akin Rice University, MEMS Dept.

Transcription

1 Computer Aided Design (CAD) Is based on parametric models Copyright 2010

2 Computer Aided Design (CAD) Creates parametric solid models for: 1. Automatic multi-view drawings with dimensionsi 2. Manufacturing control with Geometric dimensioning & tolerances (GD&T) Copyright 2010

3 Computer Aided Design (CAD) Creates parametric solid models for: 3. Rapid prototypes via 3D printing (wax, plastic, metals, casting sand) 4. Computer Numerical Controlled (CNC) machining from an initial solid block Copyright 2010

4 Computer Aided Design (CAD) Creates parametric solid model data for: 5. Mass property calculations (surface area, volume, centroid, moment of inertia) 6. Product Data Management (PDM) Copyright 2010

5 Computer Aided Design (CAD) Creates parametric solid model data to feed analysis systems such as automatic mesh generators, finite element analysis (FEA), finite volume analysis (FVA), finite difference methods (FDM), mechanism motion and kinetics studies, etc. Copyright 2010

6 CAD and Finite Element Analysis Most ME CAD applications require a FEA in one or more areas: Stress Analysis Thermal Analysis Vibrations or Structural Dynamics Computational Fluid Dynamics (CFD) Electromagnetic Analysis... Copyright 2010

7 FEA Data Reliability Geometry: generally most accurate Material: accurate if standardized Mesh: requires engineering judgement Loads: less accurate, require assumptions Restraints: least accurate, drastically effects results; several reasonable restraint cases should be studied Copyright 2010

8 Primary FEA Assumptions Model geometry Material Properties Elastic modulus, Thermal Conductivity, etc. 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) Copyright 2010

9 General 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) Natural, or Neumann BC in lower space boundary differential equation (NBC) One or the other applied at a boundary point Copyright 2010

10 General Approach for FEA, 2 Understand secondary variables (SV) obtained from the gradient of the primary variables and usually combined with the material properties p Statics: strains, stresses, failure criterion Thermal: temperature gradient,heatflux Copyright 2010

11 General Approach for FEA, 3 Understand boundary conditions (BC) 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. Copyright 2010

12 FEA Accuracy PV are most accurate at the mesh nodes. SV are least accurate at the mesh nodes. SV are most accurate at the Gauss integration points SV can be post-processed for accurate nodal values (and error estimates) Copyright 2010

13 General 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. Understand dprimary variables (PV) Stress Analysis: displacements & (maybe) rotations Thermal Analysis: temperature Fluid Flow: velocity & pressure Copyright 2010

14 General Approach for FE, 4 Understand reactions needed to maintain the Essential BC Statics: Force at given displacement Moment at given rotation (if active) Thermal: Heat flux at given temperature

15 General Approach for FE, 5 1. Estimate the solution results and gradients 2. Select an acceptable error (1 %) 3. Mesh the model 4. Load the model and apply essential BC 5. Solve the model (PV), then post-process (SV) 6. Estimate the error levels A. Unacceptable error: Adapt mesh, go to 3 B. Acceptable error: Validate the analysis

16 FE Mesh (FEM) Crude meshes that look like a part are ok for images and mass properties but not for FE analysis. Local error is proportional to product of the local mesh size (h) and the gradient of the secondary variables. PV piecewise continuous polynomials of degree p, and SV are discontinuous polynomials of degree (p-1).

17 FEA Stress Models 3-D Solid, PV: 3 displacements (no rotations), SV: 6 stresses (3 normal & 3 shear stresses) 2-D Approximations Plane Stress (σ zz = 0) PV: 2 displacements, SV: 3 stresses Plane Strain (ε zz = 0) PV: 2 displacements, SV: 3 stresses (and σ zz from Poisson s ratio) Axisymmetric ( / θ = 0) PV: 2 displacements, SV: 4 stresses

18 FEA Stress Models, 2 2-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, and bottom surfaces) Plate bending PV: normal displacement, inplane rotation vector, SV: 3 stresses (each at top, middle, and bottom surfaces)

19 FEA Stress Models, 3 1-D Approximations 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 Beams (Frames), PV: 3displacements, 3 rotations, SV: axial, bending, & shear stress Thick beam, thin beam, curved beam Pipe element, pipe elbow, pipe tee

20 Local Error The error at a (non-singular) point is the product of the element size, h, the gradient of the secondary variables, and a constant dependent d 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.

21 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. Typically we want less than 1 % error.

22 Error Estimates Quite good for elliptic problems (thermal, elasticity, ideal flow), Navier-Stokes, etc. Can predict the new mesh size needed to reach the required accuracy. Can predict needed polynomial degree. Require second post-processing pass for localized (element level) gradient smoothing.

23 Primary FEA Matrix Costs Assume sparse banded linear algebra system of E equations, with a half-bandwidth of B. Full system if B = E. Storage required, S = B * E (Mb) Solution Cost, C α B * E 2 (time) 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, C C/64 Eighth symmetry, Cyclic symmetry,...

24 Symmetry yand Anti-symmetry y Use symmetry states for the maximum accuracy at the least cost in stress and thermal problems. Cut the object with symmetry planes (or surfaces) and apply new boundary conditions (EBC or NBC) to account for the removed material.

25 Symmetry (Anti-symmetry) Requires symmetry of the geometry and material properties. p Requires symmetry (anti-symmetry) of the source terms. Requires symmetry (anti-symmetry) of the essential boundary conditions.

26 Structural Model Symmetry Zero displacement normal to surface Zero rotation vector tangent to surface Anti-symmetry Zero displacement vector tangent to surface Zero rotation ti normal lto surface

27 Thermal Model Symmetry Zero gradient normal to surface (insulated surface, zero heat flux) Anti-symmetry Average temperature on surface known

28 Local Singularities All elliptical problems have local radial gradient singularities near re-entrant corners in the domain. Radius, r u = r p f(θ) u/ r = r (p-1) f(θ) Strength, p = π/c Re-entrant, C Corner: p = 2/3, weak Crack: p = 1/2, strong u/ r as r 0

29 Stress Analysis Verification, 1 Prepare initial estimates of deflections, reactions and stresses. Eyeball check the deflected shape and the principal p stress vectors. Eyeball check the contour lines for wiggles. (OK in low stress regions.)

30 Stress 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 which almost always depend on the material properties. p

31 Stress Analysis Verification, 3 The reaction resultant forces and/or moments are equal and opposite to the actual applied loading. For pressures or tractions remember to compare their integral (resultant) to the solution reactions. Reactions can be obtained at elements too.

32 Stress Analysis Verification, 4 Compare displacements, reactions and stresses to initial estimates. Investigate any differences. Check maximum error estimates, if available in the code.

33 Thermal Analysis Verification, 1 Prepare initial estimates of the temperatures, reaction flux, and heat flux vectors. Eyeball check the temperature contours and the heat flux vectors. Temperature contours should be perpendicular to an insulated boundary.

34 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 properties.

35 Thermal Analysis Verification, 3 The reaction resultant nodal heat fluxes are equal and opposite to the applied heat fluxes. For distributed heat fluxes remember to compare their integral (resultant) to the solution reactions. Reactions can be obtained at elements too.

36 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.