Introduction to Finite Element Analysis using ANSYS

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1 Introduction to Finite Element Analysis using ANSYS Sasi Kumar Tippabhotla PhD Candidate Xtreme Photovoltaics (XPV) Lab EPD, SUTD Disclaimer: The material and simulations (using Ansys student version) presented in this document are made purely for teaching and sharing knowledge and NOT made for commercial use. The concepts and examples (other than author s own research) presented here were taken from publicly available references or internet. In both the cases, the original references / sources were properly acknowledged. This document is expected to be used only for personal learning / teaching purpose.

2 Area / Units Basic Idea of Finite Elements No. of Elem Element Node Circle Area Approximation Actual Area Area of Elements No. of Elements

3 Finite Element Method A numerical method to solve (partial) differential equations Gives only approximate solution Applicable to several physical domains, for ex. Structural Thermal / Fluid Electromagnetic Coupled field Discretization of the structure into small portions Elements Connecting points between elements Nodes

4 Finite Element Analysis Procedure (Structures) Pre-processing Discretization of the structure Meshing Assign element type and properties Assign material properties Apply Boundary conditions and Loads Solution Select the solver Calculate element stiffness matrices Assemble global stiffness matrix Solve for displacements, strains, stresses etc. Post-processing Display / Output displacements, strains, stresses etc. Calculate user defined parameters from the results

5 Finite Element Analysis Software Commercial / General purpose ANSYS Simple, User friendly, low cost ABAQUS Better solver, powerful for nonlinearities,high cost NASTRAN / PATRAN Used for dynamics COMSOL Multiphysics solver Altair Hypermesh / Optistruct Good for meshing Open source / Free Cauliculix Compatiable with ABAQUS GetFEM++ - Specially for Contact Analysis The list is not exhaustive, please find more in the link below

6 Finite Element Analysis - Uses New Product Design Virtual Design of Experiments (DoE) Fatigue Life / Fracture estimation Design Optimization Identification of sensor locations for testing / validation Sensitivity analysis Existing Product Feasibility of Repairs / Upgrades Product remaining life estimation Identification of maintenance intervals Failure root cause analysis Reduces new product validation / testing costs

7 Mathematical Preliminaries of FEM Differential Equation Uniform bar with one end fixed and axially loaded on the other end and body load As = * = *(du/dx) and assume f B (x) = b X x Force balance in the small element Boundary conditions Essential: Natural: This DE can be solved for exact solution, U(x) = 2x x 2 /2 but we shall find an approximate solution now Rearrange the terms Take limit, X 0 Ref: Hughes, T. J. R., 1987, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Inc. Assume a weight function, w such that Note: the weight function should satisfy the essential boundary condition Integrate over the entire body (i.e., limits) Weighted residual formulation

8 Mathematical Preliminaries of FEM Variational Formulation Assume an approximate solution, u n Weighted residual formulation Variational Form / Weak Form of the Differential Equation 0 Advantages: No second order term simple to solve numerically Symmetry Natural boundary condition is included, need not be enforced. No double differentiation requirement for displacement trail function Ref: Hughes, T. J. R., 1987, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Inc.

9 Mathematical Preliminaries of FEM Numerical Solution Uniform bar with one end fixed and axially loaded on the other end and body load Approximate (Numerical) Solution X With, A = E = L = b = P = 1 Let u n (x) = x + a sin ( x/2) Substitute in the weighted residual equation x = - a ( 2 / 4) sin ( x/2) Weigh function Galerkin Method Boundary conditions Essential: Natural: For simplification, let A = E = L = b = P = 1 Exact solution: U(x) = 2x x 2 /2 w(x) = b sin ( x/2) (satisfies essential BC.) U n (x) = x + (16/ 3 ) sin( x/2) Ref: Hughes, T. J. R., 1987, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Inc.

10 Different Types of Elements 1 D (line) Elements e k Special Purpose Elements Spring, Truss, Beam, Pipe etc. 2 D (Plane) Elements Plate, Shell, Membrane etc. n i e k n i n j n l n j Point mass, Contact, Coupling, etc. 3 D (Solid) Elements n k 3D continuum domains

11 Spring force, F Spring Element Formulation Direct Method Nodes At node i f i = -F = -k (u j u i ) = k u i k u j At node j f j = F = k (u j u i ) = -k u i + k u j Matrix Form k u = F Nodal displacements Spring displacement, u k Element stiffness matrix Element nodal displacement vector Element nodal force vector Ref: Yijun Liu, Lecture Notes Introduction to FEM, Uni. Cincinnati, 2002

Two Springs Formulation of Global Stiffness Matrix Use force equilibrium at nodes to assemble global stiffness matrix At node 1 F 1 = f 11 = k 1 u 1 k 1 u 2 At node 2 F 2 = f 21 + f 12 = -k 1 u 1 + k

12 Two Springs Formulation of Global Stiffness Matrix Use force equilibrium at nodes to assemble global stiffness matrix At node 1 F 1 = f 11 = k 1 u 1 k 1 u 2 At node 2 F 2 = f 21 + f 12 = -k 1 u 1 + k 1 u 2 + k 2 u 2 + k 2 u 3 Stiffness Matrix: Element 1 = -k 1 u 1 + (k 1 + k 2 )u 2 + k 2 u 3 At node 3 F 3 = f 22 = -k 2 u 2 + k 2 u 3 Stiffness Matrix: Element 2 Global Stiffness Matrix K U = F Ref: Yijun Liu, Lecture Notes Introduction to FEM, Uni. Cincinnati, 2002

13 Force, F Elastic Bar Element Formulation Nodal displacements At node i f i = -F = -k (u j u i ) = k u i k u j At node j f j = F = k (u j u i ) = -k u i + k u j Matrix Form k u = F Nodes k = EA/L Displacement, u Element stiffness matrix Element nodal displacement vector Element nodal force vector Ref: Yijun Liu, Lecture Notes Introduction to FEM, Uni. Cincinnati, 2002

14 Force, F Elastic Bar Element Shape Functions Nodal displacements x i x j k u = F Nodes k = EA/L Displacement, u What about displacements within the element? Interpolate from the nodal diaplacements Let u(x) = ax +b u i = a x i + b ---- (1) u j = a x j + b ---- (2) Solving (1) and (2) a = (u j - u i )/(x j - x i ) = (u j - u i )/L b = (u j x i - u i x j )/L There fore u(x) = x(u j - u i )/L + (u j x i - u i x j )/L Rearranging terms will give u(x) = u i (x j - x)/l + u j (x - x i )/L = u i N i + u j N j N i and N j are shape functions Some Rules: N i + N j =1 At node i, N i = 1, N j = 0 At node j, N i = 0, N j = 1 Ref: Yijun Liu, Lecture Notes Introduction to FEM, Uni. Cincinnati, 2002

15 Solution Methods Direct Methods Solution time is proportional to NB 2, N is the size of the matrix and B is bandwidth Suitable for small to medium problems and slender structures (small band width) Accurate, easy to handle Requires large memory space Iterative Methods Solution time is unknown before hand Suitable for large problems and bulky structures Approximate, difficult to handle depends on initial guess Reduces memory space requirement Bandwidth Ref: Yijun Liu, Lecture Notes Introduction to FEM, Uni. Cincinnati, 2002

16 Solution Methods Direct Methods Gauss Elimination Iterative Methods Gauss-Siedel Ref: Yijun Liu, Lecture Notes Introduction to FEM, Uni. Cincinnati, 2002

17 Examples: Airbus A320 High Stress Regions?? Source:

Examples: Jet Engine Failure Due to Drone Strike Impact of a foreign object on an operating jet engine Can be catastrophic Foreign object can be: a bird, debris, snow ball, now recently drones It is

18 Examples: Jet Engine Failure Due to Drone Strike Impact of a foreign object on an operating jet engine Can be catastrophic Foreign object can be: a bird, debris, snow ball, now recently drones It is customary of engine manufacturers like GE / Rolls Royce etc. to conduct tests and FEA simulations of bird strike for engine certifications Source:

19 Example: Spanner Stress Analysis High Stress Region Source:

Example: Stresses in Silicon Solar Cells (Our

$Microdiffraction Ref: S. K. Tippabhotla et al.$ , Progress in Photovoltaics 2017 (Article in

20 Example: Stresses in Silicon Solar Cells (Our Research) Solder Joint (µsxrd) PV Mini Module (Experimental Sample) µsxrd Synchrotron X-ray Microdiffraction Ref: S. K. Tippabhotla et al., Progress in Photovoltaics 2017 (Article in press) For more details please refer:

21 Example: Predicting Fracture of Micro Beam Ref: Nagamani Jaya B et al., 2012, Philosophical Magazine, 92:25-27, , DOI: /

22 Introduction to ANSYS (Classic): Cantilever Beam Consider the beam in the figure below. It is clamped on the left side and has a point force of 8kN acting downward on the right end of the beam. The beam has a length of 4 m, width of 1 m and height of 0.2 m (cross-section is a rectangle). Additionally, the beam is composed of a material which has a Young's Modulus of 2.8x10^10 Pa. Using ANSYS, calculate the following: 1. Deformation of the beam 2. Maximum bending stress along the beam 3. Bending moment along the beam 1 m 0.2 m More

23 Step1: Preferences

24 Step1: Define Element Type

25 Step1: Define Element Key Options

26 Step1: Define Material Properties

27 Step1: Define Geometry of the Beam X 4 m 0.2 m Y

28 Step1: Define Element Size Along Beam

29 Step1: Define Element Size Across Beam 1 2 3

30 Step1: Mesh Mapped mesh by corners 1 Select the area by clicking on it (notice change 2 of colour?) Pick corners in cyclic order

31 Step1: Mesh Define Analysis Type

32 Step1: Mesh Define BC 1 2 3

33 Step1: Mesh Define Load 1 2

34 Step1: Mesh Solution

35 Step1: Results Displacement, uy

36 Step1: Results Stress, Sxx

37 Mesh Refinement: Spanner Stress Analysis High Stress Region Source:

38 Exercise Stress concentration in Plate with a Hole 6 m Eqv. Stress 1.5 m 4 m Radius 500 mm 1 kn All DOF = 0 2 m Thickness = 500 mm Material: Steel E = 200 GPa Mu = 0.3

39 Some common mistakes Inconsistent units Wrong boundary conditions Wrong material property assignment Wrong element type assignment

40 Thank You

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