Analysis of Contact Lifting Devices for Cylindrical Members Using Adaptive Mesh Refinement in Two Dimensions Judy Kim – Mechanical Engineering Undergraduate ’14 Maegan Porpora – Mechanical Engineering Undergraduate ‘14 Adaptive Refinement Original Mesh h-Adaptivity p-Adaptivity Original elements are divided to create more elements Original elements are interpolated between with higher order shape functions hp-Adaptivity Real Life Application: Lifting Devices Cylindrical parts in the jet engine industry Lifting Force Simplified model of a low pressure turbine interfaced with a lifting device Our Design Loads π€πππβπ‘ 2 π€πππβπ‘ 2 F 3D Representation with a singularity for the lifting attachment Simplified 2D approximation with a distributed load for the lifting attachment and point forces at the ends to represent contact force Purpose • Compare h and p adaptivity • Which one requires fewer refinements to meet a target error? • Which one is easier to implement? • Is analysis easier in MATLAB or ANSYS? • Are the results consistent between the two softwares? Outline • Error Estimators • MATLAB • ANSYS • 3 New Geometries • Conclusions • Future Recommendations Initial Problem Setup Weak form for 2D linear elasticity πππ π€π πΏπ π π Ωπ π=1 π π΅ π π·π π΅π πΩπΏπ π − π π π π‘πΓ − Γπ π π ππΩ Ωπ = 0 ∀ π€πΉ ∈ π0 There are no body forces in our problem, so this reduces to: πππ π€π πΏπ π π=1 πΎπ = π π π π‘πΓ Ωπ Γπ π ππ = π΅ π π·π π΅π πΩ Ωπ π π΅ π π·π π΅π πΩπΏπ π − = 0 ∀ π€πΉ ∈ π0 π π π π‘πΓ Γπ Summary of A Posteriori Error Estimation “From what comes later” Construct error estimators from local info to judge accuracy of solution Types of a posteriori error estimators: • Explicit • Implicit • Recovery-Based Zienkiewicz-Zhu Patch Recovery Technique 1/2 1 1D Example πΊ − π»πβ 2 πΩ ππ = πΊ − π»πβ = 0 π»πβ = π΅π {π π } πΊ = ππ πΊπ Discontinu ous ππ = πππππ ππ πππππππ‘ πΎ π»πβ = ππππππππ‘ ππ πΉπΈ π πππ’π‘πππ πΊ = πππππ£ππππ ππππππππ‘ ππ πΉπΈ π πππ’π‘πππ Piecewise Continuous [1] Zienkiewicz-Zhu Error Estimator Algorithm π Calculate FE solution Calculate recovered gradient G For each element For all 4 Gauss Points Calculate N Calculate B Calculate J and |J| Interpolate G to Gauss Point Interpolate π»g to Gauss Point Calculate error by adding weighted result at each Gauss Point, take square root (− 1 1 , ) 3 3 ( 1 1 , ) 3 3 π (− 1 1 , − ) 3 3 π1π π= 0 π΅= ππ1π ππ₯ 0 0 π2π π1π 0 0 ππ1π ππ¦ ππ2π ππ₯ 0 ( 1 1 , − ) 3 3 0 π3π π2π 0 0 ππ2π ππ¦ ππ3π ππ₯ 0 0 π4π π3π 0 0 ππ3π ππ¦ 0 π4π ππ4π ππ₯ 0 0 ππ4π ππ¦ Zienkiewicz-Zhu Error Estimator Algorithm Calculate FE solution Calculate recovered gradient G For each element For all 4 Gauss Points Calculate N Calculate B Calculate J and |J| Interpolate G to Gauss Point Interpolate π»g to Gauss Point Calculate error by adding weighted result at each Gauss Point, take square root Sum up local error If total error < πΎ π’β For each element Mark for refinement ππ = πππππ πππππ Σππ2 = π‘ππ‘ππ πππππ π’β = ππππππ ππππππ¦ ππππ = π π πΎπ πΎ = πππππ π‘ππππππππ (0 < πΎ < 1) πππ = ππ’ππππ ππ πππππππ‘π MATLAB Analysis with hAdaptivity πΎ = 10% Iterati on: 1 2 3 MATLAB Analysis with pAdaptivity 1st Order 2nd Order 3rd Order ANSYS h-Adaptivity 4 Noded Quadrilateral Elements were used Mesh size began at a scale of 10 The lifting device boundaries were divided as shown ANSYS p-Adaptivity Plane182 – 4 Noded Quadrilateral Plane183 – 8 Noded Quadrilateral Note: Plane145 or “p-method” were discontinued with ANSYS 13.0 but used to allow the user to implement shape functions up to polynomial order eight Verifying Results with ANSYS andOriginal MATLAB Setup Refined Mesh (2nd order shape functions) • Zero displacement on side 5 • Loads of -111N in the Y direction on the highlighted nodes • MATLAB uses 9 nodes per element • ANSYS uses 8 nodes per element Program Max xdisplace Max ydisplace Program Max xdisplace Max ydisplace MATLAB -4.3265e-8 -1.1511e-8 MATLAB -3.72e-8 -1.32e-8 ANSYS -4.0e-8 -1.06e-8 ANSYS -4.48e-8 -1.23e-8 Error 8% 9% Error 7% 17% Deformation Plots ANSYS MATLAB Shared Errors ANSYS MATLAB Additional Geometries • Shaped to mirror the deformation curve • Minimizes material Max Deformation 7.702e-7 Max Stress 0.570 MPa Max Error at last refinement 1.628e-7 Total Iterations 6 Additional Geometries • Truss shaped geometry • Minimizes material while utilizing the strength of a truss shaped structure Max Deformation 1.409e-7 Max Stress 0.3613 MPa Max Error at last refinement 5.614e-8 Total Iterations 6 Additional Geometries • Rectangular geometry with holes through the thickness • Minimizes material Max Deformation 7.501e-8 m Max Stress 0.105 MPa Max Error at last refinement Total Iterations 5.589e-8 2 Conclusions • ANSYS analysis is much easier to implement than creating a code through MATLAB • Consistent results can be achieved through both softwares • Depending on the geometry, the mesh will converge at different rates under the same loading conditions • All geometries analyzed provide an adequate factor of safety Recommendations for Future Analysis • Automated MATLAB refinement • Bringing back p-method ANSYS elements • Ability to choose element type for individual elements in ANSYS References [1] Ainsworth, Mark, and J. 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