1. Thermoelastic analysis with a home-made FEM Tübingen 8^th-9^th 20074th ILIAS-GW Annual General Meeting Michele Bonaldi, Enrico Serra

2. Presentation overview Why we develop our own FE formulation for coupled thermoelasticity and dissipation ? Code validation: the problem of a flexure simply- supported beam Code developments: model order reduction techniques

3. Why we started to develop an Finite Element code ?

4. Home made FEM Objectives: - Computation of the thermoelastic dissipation for “critical” materials - Computation of the thermoelastic dissipation for thin layers over substrate (bonding - coating) Advantages: - Linking FE solver with Mathematica Arbitrary Precision routines - A full control on the FE equations (compute Entropy)‏ - Freedom on using specialized solution algorithm

5. How useful is arbitrary-precision (AP) in FE formulations By using AP, numerical errors present in the discretization and solution phase of a physical problem can be put under control (in our case we use AP only in the solution phase). The effect of the precision can be well understood when comparing Double Precision and AP results with analytical predictions. A random walk path of the solution error is present for DB result when the number of elements increases. Physical system Mathematical model Discrete Model Discrete solution IDEALIZATION DISCRETIZATION SOLUTION (FEM,..)‏ Modelling + discretization + solution errors

6. The FE formulation for TED (1)‏ Starting from the principle of virtual work for dynamic thermoelasticity [Biot]: By using linear constitutive equations, the unknowns becomes the temperature difference and the displacement : Fourier's lawEntropy constitutive eq. Stress – strain, temperature eq.

7. The FE formulation for TED (2) The entropy produced for unit volume due to irreversible heat conduction must be computed as: The dissipation angle is computed as the ratio between the work lost per cycle and the maximum elastic energy during the cycle: After discretization via 20-nodes iso-parametric Serendipity shape functions the following matrix system is obtained (in the harmonic case): An Entropy approach [Landau, Lifshitz] NOT AVAILABLE IN COMMERCIAL FEM is followed for computing the dissipation angle

8. APFEM concepts and performances Ansys or others FE commercial codes can be used for mesh generation -> APFEM_ini.m | | -> APFEM_mesh.m | | | -> APFEM_solver.m | -> APFEM_aux.m FE engine is developed by using Mathematica primitives. Our attempt was to improve computational time and memory management according to the limitations of a not compiled code A user interface dialogue menu was developed

9. Case Study: TED in a flexure simply supported beam A TED problem for a simply supported flexure beam was considered. The temperature profile along the beam's width can be calculated using [Lifshitz-Roukes] formula. Only with this kind of boundary conditions the curvature of the neutral axis can be expressed in closed-form. Using L-R formula the temperature profile becomes: Results agree (320 elements )‏ within 8.5 % Adiabatic conditions on all surfaces

10. Convergence check The convergence problem for a 10 mm x10 mm rectangular cross-section FuSi beam is still present although we use a 25 digit-precision arithmetic. On the other hand, we used a different element integration formula in the FE scheme and the Entropy approach with respect to ANSYS but results agree within 2%

11. GOOD new: APFEM works correctly. BAD new: arbitrary precisions is not sufficient to solve convergence problems

12. We can now give an answer to the convergence problem Question: ill-conditioned matrices affect the final solution ? Answer: No. In fact we find out that in the critical case the matrix condition number is lower than those evaluated in the other cases. APFEM allowed the used of adimensional equations (with better conditioning) but solution doesn't change. Parabolic thermal eq. gives rise to the convergence problem: The propagation constant and the penetration depth for the thermal field Thermal diffusion along each side of the element Matrix condition numbers (the best is the lowest)‏ For 10 mm FuSi beam, 869 elements along beam's width are needed in order to have an equal discretization level of a 92um Si beam.

13. Model order reduction (MOR) for thermoelasticity problems (1)‏ MOR methods derived from the control theory are well suited for linear thermoelasticity. We are trying to use a model order reduction strategy in conjunction with APFEM in order to address large-scale problems where a finer mesh grid is needed (FuSi simply supported beam, bonded-coated masses,... ). By using those methods a large-scale model is replaced by an approximate smaller model, which is capable of capturing dynamical behaviour and preserving essential properties of the larger one. = = We are dealing with second-order dynamical system and we use the most promising method based on Krylov subspaces with the moment-matching property satisfied: Transfer matrix At least 2n moments match ! Taylor Expansion

14. Model order reduction for thermoelasticity problems (2)‏ We follow the second-order Krylov's subspace approach using one-side home-made Arnoldi algorithm for constructing the matrix projection base. In building Krylov's subspace base, matrix products and a matrix inversion is required. We find out a solution strategy in case of low-frequency harmonic thermoelastic analysis that performs matrix inversion of two lower dimension positive definite real symmetric matrixes. Compared to the original problem the real advantage in using MOR is inverting two positive definite real symmetric matrixes instead of solving a large scale complex unsymmetric system. Relative error on the dissipation angle by using APFEM (with 25-precision arithmetic) Solving a 10 X 10 system give rise to a 10^-5 relative percentage error. The full system dimension is 21000X21000 !

15. Conclusions and further developments An Abitrary Precision FE code is developed by using Mathematica sw packages and can be applied to solve several kind of linear thermoelastic problems (mutilayered structures, etc )‏ Large system of equations can be conveniently treated with advanced MOR techniques By using the finite element engine other approaches (analytical, meshless) can be linked to it for treating bonded stuctures in an efficiently way. By using the finite element engine we can also deal with generalized thermoelasticity where the thermal wave speed becomes finite (mems and nems).