Presentation on theme: "By: Rachel Sorna and William weinlandt"— Presentation transcript:
1 By: Rachel Sorna and William weinlandt 3D Finite Element Modeling: A comparison of common element types and patch test verificationBy: Rachel Sorna and William weinlandt
2 ObjectivesDevelop a sound understanding of 3D stress analysis through derivation, construction, and implementation of our own 3D FEM Matlab Code.Compare the accuracy of two different element types mentioned in classLinear TetrahedralTri-Linear HexahedralUnderstand and utilize the patch test as a means of verifying our code and element modeling
3 3D FEM Code Attempt #1 – Conversion from 2DStressAnalysis Code Can you guess what loading scheme created the deformation on the right?Answer: Uniform traction of 1000N/m2 along the top boundary…
5 Major Parts of Code Construction Addition of third dimensional variable, zetaManual generation of meshes and defining elements and corresponding nodes3D Elasticity matrix3D Transformation matrixDefining solid elements and corresponding boundary surface elementsShape Functions (N matrix)Shape Function Derivatives (B Matrix)Gauss points and weightsDetermining which element face was on a given boundary
6 Linear TetrahedralSolid Volume ElementBoundary Surface Element
7 Tri-Linear Hexahedral Solid Volume ElementBoundary Surface Element
8 Patch Test: Code and Element Verification Two simple, yet effective tests were done: fixed displacement and fixed tractionTest Subject: A simple CubeMaterial Properties:E: 200GPaν: .3Block Dimensions:Height: 5mWidth: 5mDepth: 5mHexahedral MeshingTetrahedral Meshing
9 Fixed Displacement Test 𝜎= 𝛿 𝐿 𝐸Fixed displacement of was applied in the negative Z-direction. The following analytical solution gives a stress value of 4MPa throughout the cube:Table 1. Normalized L2 Norm Error values for tetrahedral and hexahedral elements for varying degrees of mesh fineness.Element TypeMesh FinenessTetrahedralHexahedralCoarse(48/64)3.6e-37.56e-4Moderate(750/729) 1.06e-3 2.50e-5Fine(3000/3375) 3.06e-4 6.25e-6
10 Tri-Linear Hexahedral Fixed Traction Test𝛿= 𝐹𝐿 𝐴𝐸Fixed traction of 1000N/m2 was applied in the positive Z-direction. The following analytical solution gives a displacement value of 2.5e-8m on the top surface:Table 2. Normalized L2 Norm Error values for tetrahedral and hexahedral elements for varying degrees of mesh fineness.Element TypeMesh FinenessLinear TetrahedralTri-Linear HexahedralCoarse(24/27 )1.06e-23.69e-4Moderate(750/729)1.90e-34.62e-5Fine(3000/3375) 3.60e-5 1.60e-5
11 Application – A Complex Loading Scheme Counterclockwise oriented 1000N shearing surface tractions applied to the X and Y faces produced the following ‘twisty’ cube. Comparison to the same loading scheme in ANSYS revealed that our model was valid for complex loading schemes as well.Our CodeANSYS
12 Convergence PlotThe following plot shows the convergence of von-Mises stress for tetrahedral and hexahedral elements performed using our code as well as ANSYS for the ‘twisty’ cube loading condition.
13 Current Limitations of Our Code Matlab can only handle so many elements and thus degrees of freedom before it becomes impossibly slow or runs out of memoryLimitations of the variation in orientation of tetrahedral elementsLimited possible geometriesLinear elements only, no quadratic elements
14 ConclusionsHexahedral elements appear to more accurately model simple linear deformation problems than tetrahedral elements – they also take far less time!Tetrahedral elements can be useful for complex geometries and loading conditions (rounded surfaces, sharp corners, etc.)For accurate meshing and modeling of complex multi-part geometries, mixed meshing, or using both tetrahedral and hexahedral, as well as other element types, is best!Writing your own FEM Code is a long and arduous process, but once you are done, you have an unparalelled understanding of the fundamental concepts!