Physics-informed neural networks for the analysis and optimization of structures MAI TIEN HAU February 2023 Department of Architectural Engineering The Graduate School Sejong University Physics-informed neural networks for the analysis and optimization of structures MAI TIEN HAU A dissertation submitted to Faculty of Sejong University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Architectural Engineering February 2023 Approved by Professor Jaehong Lee Major Advisor Physics-informed neural networks for the analysis and optimization of structures by MAI TIEN HAU Approved------------------------------------------------------------------------- Professor Kihak Lee, Chair of the committee Approved------------------------------------------------------------------------- Professor Dongkyu Lee, Member of dissertation committee Approved------------------------------------------------------------------------- Professor Seunghye Lee, Member of dissertation committee Approved------------------------------------------------------------------------- Professor JongJae Lee, Member of dissertation committee Approved------------------------------------------------------------------------- Professor Jaehong Lee, Advisor ABSTRACT This thesis is concerned with nonlinear, stability analyses, and size optimization of truss structures based on physics-informed neural networks (PINNs). For non linear analysis one, a robust and simple unsupervised neural network framework is proposed to perform the geometrically nonlinear analysis of inelastic truss struc tures. To guide the training process, the loss function built via the total potential energy principle under boundary conditions (BCs) is minimized in the suggested NN model whose weights and biases are considered as design variables. And the training data only contain the spatial coordinates of joints.
In each training iter ation, feedforward, physical laws, and back-propagation are applied for adjusting the parameters of the network to minimize the loss function. Once the network is properly trained, the mechanical responses of inelastic structures can be eas ily obtained without using any structural analysis as well as incremental-iterative algorithms. Several benchmark examples regarding geometrical and material non linear analysis of truss structures are examined to demonstrate the effectiveness and reliability of the proposed paradigm. Subsequently, the proposed model is first to analyze the stability of truss structures.
Different from most existing work, neural network (NN) is designed to directly locate the critical point by minimizing the loss function involving the residual load and property of the stiff i ness matrix which they are established based on the outputs, loads, and BCs. It is also significant because the first critical point will be located at the training end corresponding to the minimum loss function without utilizing any incremental- iterative methods. Additionally, this dissertation also develops a Bayesian deep neural network-based parameterization framework to directly solve the optimum design for geometrically nonlinear trusses for the first time. In this approach, the parameters of the network are regarded as decision variables of the structural op timization problem, instead of the member’s cross-sectional areas.
Therein, the loss function is constructed with the aim of minimizing the total structure weight so that all constraints of the optimization problem obtained by supporting fi nite element analysis (FEA) and arc-length method are satisfied. Furthermore, Bayesian optimization (BO) is applied to automatically tune the hyperparame ters of the network. The effectiveness of this model is demonstrated through a series of numerical examples for geometrically nonlinear space trusses. And the obtained results demonstrate that our framework can overcome the drawbacks of applications of machine learning in computational mechanics.
Finally, a physics- informed neural energy-force network (PINEFN) framework is first constructed to directly solve the optimum design of truss structures that structural analy sis is completely removed from the implementation of the global optimization in this thesis. Herein, the loss function is designed based on the output values and physics laws to guide the training. Now only NN is used in our scheme to minimize the loss function wherein weights and biases of the network are con sidered as design variables. In this model, spatial coordinates of truss members are examined as input data, while corresponding cross-sectional areas and re dundant forces unknown to the network are taken account of output.
Obtained outcomes indicated that it not only reduces the computational cost dramatically ii but also yields higher accuracy and faster convergence speed compared with re cent literature. With the above outstanding features, it is promising to offer a unified solver-free numerical simulation for solving complex issues in structural optimization. Keywords: Physics-informed, Geometric nonlinear, Structural stabil ity, Hyperparameter optimization, Force method, Critical points, Com plementary energy, Bayesian optimization, Truss optimization iii CONTENTS ABSTRACT. i LIST OF TABLES.
vii LIST OF FIGURES.3 Physics-informed neural networks. 9 2 PHYSICS-INFORMED NEURAL NETWORK FOR NON LIN EAR ANALYSIS OF TRUSS STRUCTURES.2 PINN for nonlinear analysis.2 Unsupervised learning-basedapproachframework.3 PINN for structural stability analysis.2 Direct instability-informed neuralnetworkframework .1 Material and geometrical nonlinearities. 56 3 BAYESIAN DEEP NEUARL NETWORK-BASED PARAM ETERIZATION FRAMEWORK FOR OPTIMUM DESIGN OF NONLINEAR STRUCTURES.2 Statement of structuraloptimizationproblem.3 BDNN-based parameterizationframework.1 DNN-based parameterization model.1 25-bar space truss.2 52-bar dome truss.3 56-bar space truss.4 120-bar dome truss. 89 4 PHYSICS-INFORMED NEURAL ENERGY FORCE NET WOR K FOR STRUCTURAL OPTIMIZATION.2 Structural optimization basedon energy-force methods.3 Physics-informed neuralenergy-force network.1 Ten-bar truss.2 200-bar planar truss.3 25-bar space truss.4 72-bar space truss.5 120-bar dome truss.
130 5 CONCLUSIONS AND FUTURE WORK. 135 LIST OF PUBLICATIONS. 150 ABSTRACT IN KOREAN. 155 VI LIST OF TABLES 2.1 Type of material, network architecture, and epoch for different problems.2 Comparison of joint displacements and energy for 6-bars truss with different solution techniques and different materials.3 Comparison of member forces for 6-bars truss with different solu tion techniques and different materials.4 Comparision of joint displacement results for the 31-bars truss with different algorithms and various materials.5 Comparision of member forces for the 31-bars truss with different algorithms and various materials.6 Loading conditions for the 25-bar space truss with geometrical nonlinearity.7 Comparison of displacements for 25-bars truss with various loading conditions.8 Comparison of member forces for 25-bars truss with various loading conditions.9 Comparison of error percentage of different algorithms with FEM for the 25-bar truss.10 Comparison of deflection results for the 52-bar dome truss obtained by different algorithms.11 Comparison of member forces for 52-bars truss obtained by differ ent algorithms.12 Comparison of member forces for the 10-bar truss with different algorithms and various materials.13 Comparison of error percentage of different algorithms with ILM for the 10-bar truss.14 Comparison of member forces for the 21-bar truss with different algorithms and various materials.15 Comparison of error percentage of different algorithms with ILM for the 21-bar truss.16 Comparison results obtained for the two-bar truss in searching the first snap-through point with Ku=0.17 Comparison results obtained for the two-bar truss in searching the first snap-through point with varying Ku.18 Comparison of results of the triangular truss dome for searching the bifurcation point.19 Loading conditions for dome truss with 24 bars.20 Comparison results obtained for 24-bar dome truss in searching the first critical point.1 Configuration space for the hyperparameters of the network .2 Material properties, upper and lower bounds on design variables.3 Loading condition for the 25-bar space truss.4 Optimum hyperparameters of the network obtained using the BO with different acquisition functions for the 25-bar space truss.5 Statistics of the optimal weight with different acquisition functions for the 25-bar space truss.6 Comparison of optimal results for the 25-bar space truss.7 The displacement constraints of the 25-bar space truss.8 Optimum hyperparameters obtained by using the BO for different problems.9 Statistics of the optimal weight with different problems.10 Comparison of optimal results for the 52-bar dome truss.11 The displacement constraints of the 52-bar dome truss.12 Comparison of optimal results for the 56-bar space truss.13 The displacement constraints of the 56-bar space truss.14 Comparison of optimal results of the 120 dome truss.15 The displacement constraints of the 120-bar dome truss.1 Hyperparameters for the benchmarks tested in this study.2 Comparison of the obtained results for the 10-bar truss with the first loading condition.3 Error of the constraints for the 10-bar planar truss with the first loading condition.4 Comparison of the obtained results for 10-bar planar truss with the second loading condition.5 Error of the constraints for the 10-bar planar truss with the second loading condition.6 Design variables of the 200-bar planar truss.7 Optimization results obtained for the 200-bar planar truss.8 Error of the constraints for the 200-bar planar truss.9 Loading conditions for the 25-bar space truss (kips).10 Stress limitation for the 25-bar space truss.11 Optimization results obtained for the 25-bar space truss.12 Error of the constraints for the 25-bar space truss.13 Loading conditions for the 72-bar space truss (kips).14 Optimization results obtained for the 72-bar space truss (Case 01).15 Error of the constraints for the 72-bar planar truss (Case 01).16 Optimization results obtained for the 72-bar space truss (Case 02).17 Error of the constraints for the 72-bar space truss (Case 02).18 Optimization results obtained for the 120-bar dome truss.19 Error of the constraints for the 120-bar dome truss.
130 X LIST OF FIGURES 1.1 Flowchart depicting the data-driven approach for structural analysis.2 The schematic process of PINN for linear elasticity problem.1 Deformation of a space truss element.2 The whole process of an unsupervised learning-based framework for geometrically nonlinear analysis of inelastic truss structures.3 Schematic representation of snap-through and bifurcation points.4 Indirect approach using bi-section method.5 The whole process of the direct instability-informed neural network framework.6 Stress—strain relationships considered in the analysis.7 A 6-bar planar truss structure.8 The convergence histories of the loss function for the 6-bar truss with the materials MAT3 and MAT4.9 A 31-bar planar truss structure.10 The convergence histories of the loss function for the 31-bar truss with different materials.11 Schematic of a 25-bar space truss.12 52-bar dome space truss structure.13 The convergence history of the loss function for the 52-bar dome truss.14 A 10-bar planar truss structure.15 21-bar two-span continuous truss structure.16 Two-bar planar truss.17 The loss convergence history of the two-bar truss with Ku = 0.18 Load-deflection curve for the two-bar truss with Ku = 0.19 Effect of the spring stiffness on the structural stability.20 Triangular truss dome.21 Load-deflection curve and critical points of the triangular truss dome 51 2.22 Star dome truss with 24 bars.23 Case 01: Load-deflection curve and critical points of the 24-bar dome truss.24 Case 02: Load-deflection curve and critical points of the 24-bar dome truss.25 Case 03: Load-deflection curve and critical points of the 24-bar dome truss.1 Schematic of integration of BDNN-based parameterization frame work for structural optimization .2 An example of EI based Bayesian optimization of a one-dimensional minimization problem.3 A 25-bar space truss structure.4 The convergence histories of the HPO using BO for the 25-bar space truss structure.5 Iteration history of the SQP algorithm for different initial areas for the 25-bar space truss.6 The weight convergence histories of the optimal network and other studies for the 25-bar space truss.7 Schematic of a 52-bar dome truss structure.8 The convergence history of the HPO using BO for the 52-bar dome truss structure.9 The weight convergence histories of the optimal network and other works for the 52-bar dome truss.10 A 56-bar space truss structure.11 The convergence history of the HPO using BO for the 56-bar space truss structure.12 The weight convergence histories of the optimal network and FEA- DE for the 56-bar space truss.13 120-bar dome space truss structure.14 The convergence histories of the HPO using BO for the 120-bar dome truss structure.15 The weight convergence histories of the optimal network and FEA- DE for the 120-bar dome truss.1 Process of structural optimization, (a) Conventional approach in cluding optimizer algorithm and structural analysis, (b) Framework combines between the deep neural network and structural analysis. (c) Physics-informed neural network without using any structural analyses.2 Physics-informed neural energy-force networks framework for de sign optimization.3 A 10-bar planar truss structure.4 The weight convergence histories of the 10-bar truss obtained using the PINEFN and DE for the first load case.5 The weight convergence histories of the 10-bar truss obtained using the PINEFN and DE for the second load case.6 A 200-bar planar truss structure.7 The weight convergence histories of the 200-bar truss obtained using the PINEFN and other algorithms.8 A 25-bar space truss structure.9 The weight convergence histories of the 25-bar truss obtained using the PINEFN and other algorithms.10 A 72-bar space truss structure.11 The weight convergence histories of the 72-bar truss obtained using the PINEFN and other algorithms for the first load case.