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MANE 4240 & CIVL 4240 Introduction to Finite Elements Convergence of analysis results Prof. Suvranu De

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Reading assignment: Lecture notes Summary: Concept of convergence Criteria for monotonic convergence : completeness (rigid body modes + constant strain) + compatibility Incompatible elements and the patch test Rate of convergence

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Errors that affect finite element solution results Type of errorSource 1. Discretization errorUse of FE interpolations for geometry and solution variables 2. Numerical integrationEvaluation of FE element matrices and vectors using numerical integration 3. Round offThis error is due to the finite precision arithmetic used in digital computers

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What is “convergence”? Physical system Mathematical model FE model “Convergence” of FE solution results to the exact solution of the mathematical model FE scheme exhibits convergence if the Discretization error →0 as the mesh is made infinitely fine (i.e., element size →0)

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Mesh refinement h-refinement p-refinement h=element size p=polynomial order

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Convergence in energy and displacement u : exact displacement solution to a problem that makes the potential energy of the system a minimum corresponding stress and strain Exact strain energy of the body u h : FE solution (‘h’ refers to the element size) corresponding stress and strain Approximate strain energy of the body

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Calculation of strain energies Example: 80cm 1 2 Consider a linear elastic bar with varying cross section x The governing differential (equilibrium) equation Boundary conditions Analytical solution Eq(1) E: Young’s modulus P=3E/80

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The exact strain energy of the system is If we discretize the problem using a single linear finite element, the stiffness matrix is The strain energy of the FE system is Note

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Convergence in strain energy Monotonic convergence Nonmonotonic convergence

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Convergence in displacement Monotonic convergence Nonmonotonic convergence

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Criteria for monotonic convergence 1. COMPLETENESS 2. COMPATIBILITY © 2002 Brooks/Cole Publishing / Thomson Learning™

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CONDITION 1. COMPLETENESS This requires that the displacement interpolation functions must be chosen so that the elements can represent 1. Rigid body modes 2. Constant strain states

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Rigid body modes

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The # of rigid body modes of an element = # of zero eigenvalues of the element stiffness matrix

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Constant strain states x Actual variation of strain Strain computed using linear finite elements

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Mathematical implication of the two conditions (rigid body modes + constant strain state) Inside a finite element (of any order) in 1D but this is just a polynomial… Hence The requirement for completeness in 1D is that the displacement approximation be at least a linear polynomial of degree (k=1), ie any 2 node element and higher is complete

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Mathematical implication of the two conditions (rigid body modes + constant strain state) Inside a finite element (of any order) in 2D but this is just a polynomial… Hence The requirement for completeness in 1D is that the displacement approximation be at least a linear polynomial of degree (k=1).

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Mathematical implication of the two conditions (rigid body modes + constant strain state) The element displacement approximation must be at least a COMPLETE polynomial of degree one 1D 2D k=1

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In 2D, the minimum displacement assumption needs to be Translation along x Translation along y Rigid body rotation about z-axis

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CONDITION 2. COMPATIBILITY The assumed displacement variations are continuous within elements and across inter-element boundaries Ensures that strains are bounded within elements and across element boundaries. If ‘u’ is discontinuous across element boundaries then the strains blow up in-between elements and this leads to erroneous contributions to the potential energy of the structure Physical meaning: no gaps/cracks open up when the finite element assemblage is loaded

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Nonconforming elements and the patch test Conforming = compatible Nonconforming = incompatible Ideal: Conforming elements Observation: Certain nonconforming elements also give good results, at the expense of nonmonotonic convergence Nonconforming elements: satisfy completeness do not satisfy compatibility result in at least nonmonotonic convergence if the element assemblage as a whole is complete, i.e., they satisfy the PATCH TEST

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PATCH TEST: 1. A patch of elements is subjected to the minimum displacement boundary conditions to eliminate all rigid body motions 2. Apply to boundary nodal points forces or displacements which should result in a state of constant stress within the assemblage 3. Nodes not on the boundary are neither loaded nor restrained. 4. Compute the displacements of nodes which do not have a prescribed value 5. Compute the stresses and strains The patch test is passed if the computed stresses and strains match the expected values to the limit of computer precision.

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NOTES: 1. This is a great way to debug a computer code 2. Conforming elements ALWAYS pass the patch test 3. Nodes not on the boundary are neither loaded nor restrained. 4. Since a patch may also consist of a single element, this test may be used to check the completeness of a single element 5. The number of constant stress states in a patch test depends on the actual number of constant stress states in the mathematical model (3 for plane stress analysis. 6 for a full 3D analysis)

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CONVERGENCE RATE This is a measure of how fast the discretization error goes to zero a the mesh is refined Convergence rate depends on the order of the complete polynomial (k) used in the displacement approximation k=1 k=2 k=3

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It can be shown that for (1) a sufficiently refined mesh and (2) for problems whose analytical solution does not contain singularities Convergence in strain energy : order 2k Convergence in displacements : order p=k+1 C and C 1 are constants independent of ‘h’ but dependent on 1. the analytical solution 2. material properties 3. type of element used

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slope = 2 Large C shifts curve up Ex: for a domain discretized using 4 node plane stress/strain elements (k=1)

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Important property of finite element solution: When the conditions of monotonic convergence are satisfied (compatibility and completeness) the finite element strain energy always underestimates the strain energy of the actual structure Strain energy of mathematical model Strain energy of FE model

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