introduction to finite element analysis using ansys to finite element analysis using ansys sasi...
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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.
Basic Idea of Finite Elements
Element
Node
1 2 6
8 12
40
50
60
70
80
90
100
110
120
0 2 4 6 8 10 12 14 16
Are
a /
Un
its
No. of Elements
Circle Area Approximation
Actual Area
Area of Elements
No. of Elem
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
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
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• https://en.wikipedia.org/wiki/List_of_finite_element_software_packages
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
Mathematical Preliminaries of FEM – Differential Equation
X
x
Rearrange the terms
Take limit, X 0
Uniform bar with one end fixed and axially loaded on the other end and body load
Force balance in the small element
As =* = *(du/dx) and assume fB(x) = b
Boundary conditions This DE can be solved for exact solution, U(x) = 2x – x2/2but we shall find an approximate solution now
Assume a weight function, w such that
Integrate over the entire body (i.e., limits)
Ref: Hughes, T. J. R., 1987, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Inc.
Weighted residual formulation
Note: the weight function should satisfy the essential boundary condition
Essential:
Natural:
Mathematical Preliminaries of FEM – Variational Formulation
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
Assume an approximate solution, un
Ref: Hughes, T. J. R., 1987, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Inc.
Weighted residual formulation
0
Variational Form / Weak Form of the Differential Equation
Mathematical Preliminaries of FEM – Numerical Solution
X
x
Uniform bar with one end fixed and axially loaded on the other end and body load
Ref: Hughes, T. J. R., 1987, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Inc.
Boundary conditions
Essential:
Natural:Exact solution: U(x) = 2x – x2/2
With, A = E = L = b = P = 1Let un(x) = x + a sin (x/2)
Weigh function – Galerkin Method
= - a (2 / 4) sin (x/2)
w(x) = b sin (x/2) Un(x) = x + (16/3) sin(x/2)
Approximate (Numerical) Solution
(satisfies essential BC.)
Substitute in the weighted residual equation
For simplification, let A = E = L = b = P = 1
Different Types of Elements
1 D (line) Elements
Spring, Truss, Beam, Pipe etc.ni
nj
ek
2 D (Plane) Elements
Plate, Shell, Membrane etc.
ni
nj
nl
nk
ek
3 D (Solid) Elements
3D continuum domains
Special Purpose Elements
Point mass, Contact, Coupling, etc.
Spring Element Formulation – Direct MethodNodes
Nodal displacements
k u = F
Matrix Form
Elementnodal displacementvector
Element nodal force vector
Element stiffness matrix
Spring displacement, u
Spri
ng
forc
e, F
k
Ref: Yijun Liu, Lecture Notes – Introduction to FEM, Uni. Cincinnati, 2002
At node ifi = -F = -k (uj – ui) = k ui – k uj
At node jfj = F = k (uj – ui) = -k ui + k uj
Two Springs – Formulation of Global Stiffness Matrix
Ref: Yijun Liu, Lecture Notes – Introduction to FEM, Uni. Cincinnati, 2002
Stiffness Matrix: Element 1
Stiffness Matrix: Element 2
Use force equilibrium at nodes to assemble global stiffness matrix
At node 1
F1 = f11 = k1 u1 – k1 u2
At node 2
F2 = f21 + f1
2 = -k1 u1 + k1 u2 + k2 u2 + k2 u3
= -k1 u1 + (k1 + k2 )u2 + k2 u3
Global Stiffness Matrix
F3 = f22 = -k2 u2 + k2 u3
At node 3
K U = F
Elastic Bar Element – FormulationNodal displacements
Nodes
k u = F
Matrix Form
Elementnodal displacementvector
Element nodal force vector
Element stiffness matrix
Displacement, u
Forc
e, F
k = EA/L
Ref: Yijun Liu, Lecture Notes – Introduction to FEM, Uni. Cincinnati, 2002
At node ifi = -F = -k (uj – ui) = k ui – k uj
At node jfj = F = k (uj – ui) = -k ui + k uj
Elastic Bar Element – Shape FunctionsNodal displacements
Nodes
k u = F
Displacement, u
Forc
e, F
k = EA/L
Ref: Yijun Liu, Lecture Notes – Introduction to FEM, Uni. Cincinnati, 2002
What about displacements within the element?Interpolate from the nodal diaplacements
Let u(x) = ax +bui = a xi + b ---- (1)uj = a xj + b ---- (2)
Solving (1) and (2)a = (uj - ui )/(xj - xi)
= (uj - ui )/Lb = (uj xi - ui xj )/L
There foreu(x) = x(uj - ui )/L + (uj xi - ui xj )/L
xi xj
Rearranging terms will giveu(x) = ui (xj - x)/L + uj (x - xi)/L
= ui Ni + uj Nj
Ni and Nj are shape functions
Some Rules:• Ni + Nj =1• At node i, Ni = 1, Nj = 0• At node j, Ni = 0, Nj = 1
Solution MethodsDirect Methods• Solution time is proportional to NB2, 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
Ref: Yijun Liu, Lecture Notes – Introduction to FEM, Uni. Cincinnati, 2002
Bandwidth
Solution Methods
Ref: Yijun Liu, Lecture Notes – Introduction to FEM, Uni. Cincinnati, 2002
Direct Methods – Gauss Elimination Iterative Methods – Gauss-Siedel
Examples: Airbus A320
Source: https://www.pinterest.com/pin/334392341058164942/
High Stress Regions ??
Examples: Jet Engine Failure Due to Drone Strike
Source: http://www.uasvision.com/2015/11/09/drone-strikes-could-cause-jet-engine-failure/
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
Example: Spanner Stress Analysis
High Stress Region
Source: https://www.comsol.com/multiphysics/mesh-refinement
Example: Stresses in Silicon Solar Cells (Our Research)
Ref: S. K. Tippabhotla et al., Progress in Photovoltaics 2017 (Article in press)
Solder Joint
PV Mini Module (Experimental Sample)
(µSXRD)
µSXRD – Synchrotron X-ray Microdiffraction
For more details please refer: http://xml.sutd.edu.sg/publications
Example: Predicting Fracture of Micro Beam
Ref: Nagamani Jaya B et al., 2012, Philosophical Magazine, 92:25-27, 3326-3345, DOI: 10.1080/14786435.2012.669068
Introduction to ANSYS (Classic): Cantilever Beam
More Examples @: https://confluence.cornell.edu/display/SIMULATION/ANSYS+Learning+Modules
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
Step1: Mesh – Mapped mesh by corners
1 2Select the area by clicking on it (notice change of colour?)
Pick corners in cyclic order
Mesh Refinement: Spanner Stress Analysis
High Stress Region
Source: https://www.comsol.com/multiphysics/mesh-refinement
Exercise – Stress concentration in Plate with a Hole
1 kN
6 m
4 m
Radius 500 mm1.5 m
2 mAll DOF = 0
Thickness = 500 mmMaterial:SteelE = 200 GPaMu = 0.3
Eqv. Stress
Some common mistakes
• Inconsistent units• Wrong boundary conditions• Wrong material property assignment• Wrong element type assignment