MAIDROC - Multidisciplinary Analysis, Inverse Design, Robust Optimization and Control Laboratory

1. Parameter identification in differential equations: A hybrid minimization based approach MAIDROC - Multidisciplinary Analysis, Inverse Design, Robust Optimization and Control Laboratory Mechanical and Materials Engineering, Florida International University, Miami, Florida George S. Dulikravich and Sohail Reddy Department of Mechanical and Materials Engineering Florida International University, Miami, Florida 33174 Marcelo J. Colaco and Helcio R.B. Orlande Department of Mechanical Engineering, UFRJ/COPPE, Rio de Janeiro DULIKRAV@FIU.EDU HTTP://MAIDROC.FIU.EDU +1 (305) 348-7016 (Thanks to our students, postdocs and visiting scientists)

2. Department of Mechanical and Materials EngineeringCollege of Engineering and ComputingFlorida International UniversityMiami, Florida, U.S.A. Largest Hispanic minority institution in the USA: 55,000 students The number one college for Doctoral Degrees in engineering and computer science in the US including Puerto Rico (tied with Berkeley, UIUC and GATech) The number one college for Masters Degrees in engineering and computer science in the US including Puerto Rico The number one college for Bachelors Degrees in engineering and computer science among all fifty states (excluding University of Puerto Rico and Polytechnic University of Puerto Rico)

3. Objectives • Use a robust and accurate hybrid optimization algorithm to determine values of unknown parameters in a partial differential equation or a system of differential equations by minimizing a properly scaled L2-norm between calculated and observed/measured values of field variables. • Demonstrate robustness, accuracy and versatility of this approach using examples of progressive complexity.

4. Single-Objective Optimization Algorithms:Griewank’s multi-extrema test function

5. Inability of Optimization Algorithms to Find the Global MinimumExample: Griewank’s function minimization Comparison: BFGS, DE, SA, PS, Hybrid optimizer

6. Hybrid Single-Objective Constrained Optimization Algorithm • Minimize a scalar objective function of a set of design variables subject to a set of equality and inequality constraint functions • Genetic Algorithm • Simulated Annealing • Simplex (Nelder-Mead) • Differential Evolution • Gradient Search (DFP, SQP) • Pschenichny-Danilin quasi-Newton

7. Flowchart of automatic switching among modules in our single-objective hybrid optimizer

8. DFP or SQP Design Variance 0 Stalls Local Minimum Bad Mutation GA or DE Nelder-Mead Lost Generation SA Population Less Fit Objective Function Variance 0 Insufficient Random Energy Constrained Hybrid OptimizationSwitching Strategy

9. Hybrid Optimization Methods • Our most recent hybrid single-objective optimizer

10. INVERSE DETERMINATION OF TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY USING STEADY SURFACE DATA ON ARBITRARY OBJECTS

13. Variation of the thermal conductivity versus temperature for various amounts of input error in boundary temperature (a) s = 0.0 oC, (b) s = 0.1 oC, (c) s = 1.0 oC, and (d) s = 5.0 oC.

14. Predicted temperature-dependence of thermal conductivity when errors were added to the boundary heat fluxes compared to the actual linear variation of k(T).

15. Variation of the thermal conductivity versus temperature for various levels of input error in temperature, (a):s = 0.0 oC, (b): s = 0.5 oC, (c):s = 1.0 oC, and (d): s = 5.0 oC. The BEM results are compared to the actual arctangent conductivity versus temperature function

16. Inverse determination of the thermal conductivity of copper in the cryogenic range. The best inverse results are shown with various levels of input error: a) s = 0.0 K, b) s = 0.1 K, and c) s = 1.0 K).

17. Inverse prediction of temperature-dependent thermal conductivity using Kirchhoff transformation . Kirchhoff’s transform converts the governing steady-state heat conduction equation into Laplace's equation,

18. Governing equations Boundary conditions at x=L

20. A version of Levenberg-Marquardt minimizer was applied for the solution of the presented parameter estimation problem (Marquardt, 1963).

23. Inverse Problem Formulation for Determining Spatial Variation of Young’s Modulus of Elasticity For simplicity, we will work with a 1.0 m x 1.0 m plate that is fixed on two ends and under two transverse point loads . The plate thickness is 0.2 m. The forward (analysis) problem can be solved with a straightforward application of the finite element method. In our case, this involved discretizing the plate with 16 four-node isoparametric elements of the shear deformable displacement formulation (Mindlin plate elements).

25. The modulus of elasticity, E, was assumed to be a bilinear function of x and y. This function was parameterized by using the modulus value at each of the four corners of a finite element as shown hereIn this inverse problem, the objective was to determine the values of {E}={E1,E2,E3,E4} such that the finite element model produces responses that match measured values. These measurements might be displacements, strains, or stresses in the interior or on the boundary of the structure.

26. This inverse problem can then be formulated as a non-linear unconstrained minimization problem of the sum of normalized least squares differences between thecomputed responses and the measured responses

27. Case 1Deflections at four nodes with the largest sensitivities with respect to the design variables {E} were selected as simulated measurement points The displacements at these four points were then used to formulate the objective function.An initial guess {E} = {5.0 MPa, 5.0 MPa, 5.0 MPa, 5.0 MPa}Correct values{E} = {2.0 MPa, 3.0 MPa, 6.0 MPa, 8.0 MPa}.

28. In Case 1, optimized design variables after 2000 iterations were 30 percent in error. Specifically, they were {E} = {2.52 MPa, 1.98 MPa, 5.88 MPa, 6.84 MPa}.

29. Case 2Measurements of reaction forces and moments on the fixed boundaries were used The reaction force, Fz, and the reaction bending moment, M, at four boundary points were taken from the forward solution as simulated measurements. Initial guess same as used in Case 1{E} = {5.0 MPa, 5.0 MPa, 5.0 MPa, 5.0 MPa}

30. In Case 2, by iteration 100, the objective function was less than 3.0e-9 The maximum error in the predicted E(x,y) was 2.0%, an order of magnitude lower than in Case 1.The resulting values of the design variables were E = {1.96 MPa, 2.94 MPa, 5.88 MPa, 7.84 MPa}

31. Inverse determination of plasticity coefficients in metallic foam filled tubes Premise • Manufacturing of metallic foams; • Application as stiffening elements; • Improved performance; • Scope for automobile industry; • How to predict FFT behaviour using simple numerical models? Goal • Obtain a reliable methodology for calibration of a simple FEM model; • Materials: Al-alloy tube and foam (both combined and separated); • Use data from 3-point bending test;

37. Inverse determination of thermal conductivity in 3-D

39. SIMULTANEOUS DETERMINATION OF • SPATIALLY VARYING • HEAT CAPACITY AND THERMAL CONDUCTIVITY • IN ARBITRARY 2-D OBJECTS • Problem Statement • Using only boundary values of a field variable or its normal derivatives on the boundary of the solid, how can the spatial distribution of the physical properties be determined throughout the arbitrarily shaped object? • Objective • Determine the spatial distribution of multiple material properties in a solid object Florida International University MAIDROC Multidisciplinary Analysis, Inverse Design, Robust Optimization and Control Laboratory

40. Mathematical Model Transient heat conduction Discretize using FEM Resulting system of ordinary differential equations Florida International University MAIDROC Multidisciplinary Analysis, Inverse Design, Robust Optimization and Control Laboratory

41. Validation of Numerical Analysis Accuracy • Example of a thermal conductivity distribution • the analytical solution for steady state heat equation is Florida International University MAIDROC Multidisciplinary Analysis, Inverse Design, Robust Optimization and Control Laboratory k(x,y) analytical T(x,y) analytical T(x,y) by COMSOL

42. Inverse Problem Methodology Minimize the difference between computed and “measured” boundary values (J functional) Function minimized using a hybrid single-objective minimization algorithm J functional computed using the response surface based on RBF to reduce computing time Florida International University MAIDROC Multidisciplinary Analysis, Inverse Design, Robust Optimization and Control Laboratory

43. Inverse Parameter Identification Methodology Utilizing a Hybrid Optimizer and Response Surfaces Florida International University MAIDROC Multidisciplinary Analysis, Inverse Design, Robust Optimization and Control Laboratory

44. An Example of Arbitrary Geometry and Boundary Conditions Florida International University MAIDROC Multidisciplinary Analysis, Inverse Design, Robust Optimization and Control Laboratory

45. Example 1: Estimation of Parameters in the Assumed Bilinear Distributions of k and Cp Florida International University MAIDROC Multidisciplinary Analysis, Inverse Design, Robust Optimization and Control Laboratory Inversely determined: thermal conductivity (left), and specific heat (right)

46. Example 2: Estimation of Parameters in the Assumed Nonlinear Distributions of k and Cp Florida International University MAIDROC Multidisciplinary Analysis, Inverse Design, Robust Optimization and Control Laboratory Inversely determined: thermal conductivity (left), and specific heat (right)

47. Example 3: Estimation of Assumed Nonlinear Distributions with Boundary Conditions Errors Measurement Noise Florida International University MAIDROC Multidisciplinary Analysis, Inverse Design, Robust Optimization and Control Laboratory

48. MAGNETIZED FIBER ORIENTATION AND CONCENTRATION CONTROL IN SOLIDIFYING COMPOSITESShort fibers (for example, 5-10 microns in diameter and 200 microns long) are vapor-coated with a thin layer of 2-3 microns of a ferromagnetic material such as nickel. The fibers will respond to the externally applied steady magnetic fields by rotating and translating so that they become locally aligned with the magnetic lines of force.

49. The objective of this work is to explore the feasibility of manufacturing specialty metal and polymer composite materials that will have specified (desired) locally directional variation of bulk physical properties such as: thermal and electrical conductivity, modulus of elasticity, thermal expansion coefficient, etc.

50. Magnetic field pattern during solidification from the right wall – Example no. 1Specified Inversely calculated