Jiun‐Shyan Chen

We study auxetic lattice structures with curved bi‐material ligaments using the finite element method. The overall Poisson's ratio and coefficient of thermal expansion of the lattices can be simultaneously tuned to be negative by adjusting their microstructural geometries and constituent material parameters. The Young's modulus of ligaments plays an important role in controlling the effective Poisson's ratio and coefficient of thermal expansion. The size and Young's modulus of the joints that connect ligaments strongly affect the effective mechanical properties. When Young's modulus of the joint is given, a larger joint size gives rise to stronger auxeticity, but less negative thermal expansion. By tuning the coefficient of thermal expansion of each constituent, the overall coefficient of thermal expansion may be negative, zero, or positive. The bicamaterial auxetic structures studied here may be designed to possess desired properties for real‐world applications when their microstructural properties are appropriately tuned.

Conventional numerical methods for dynamic problems can be computationally intensive when addressing localized features where local adaptive refinement is needed. Local adaptive refinement is tedious in meeting the regularity requirements, and the analytical enrichment functions to capture local features are often unavailable. These complexities become even more pronounced in transient problems. This work introduces a neural network-enriched Reproducing Kernel Particle Method (NN-RKPM) for solving dynamic problems based on action minimization under a symplectic space-time framework. In this approach, RKPM is employed in the background spatial discretization and approximation, and the background solution is adaptively enriched with neural network basis functions. With NN-RKPM, the dynamic problem is solved as an optimization problem, with the time domain interpolation constructed using C 0 or C 1 temporal polynomials. The neural network enrichment functions are pre-trained during the offline stage to learn specific local features through Ritz-type energy minimization. The evolution of the background RKPM and NN enriched time-dependent solution is driven by minimizing the action functional under a symplectic formulation of mechanics for a Newtonian four-space, applied to various elastodynamics problems. The proposed method offers a unified framework consistent with the classical field theory and has been shown to accurately capture the time-dependent response of mechanical systems.

We propose a novel elastic metamaterial offering a number of tunable macroscopic mechanical properties, including effective elastic moduli, overall anisotropy, mass density, and frequency band structures. The unit cell of the periodic metastructures is composed of three-dimensional pyramid-shaped inclusions, similar to the geometry of a traditional Chinese food, Zongzi. A unique feature of this configuration is that the geometry can be described by a simple algebraic equation, enabling transformation of the geometric configuration into a few variant forms upon simple coordinate scalings. We demonstrate that a number of macroscopic properties can be adjusted by geometric parameter scaling without altering the internal structure or the constituent materials. Numerical simulations are conducted to estimate the effective elastic properties. Specifically, the effective elastic moduli can be reduced by two to three orders of magnitude compared to those of the original constituent material. In addition, the tunable mechanical properties offer flexibilities to control the frequency bandgap with adjustable bandwidths. We find that the periodic array of Zongzi matastructure could generate very wide frequency bandgaps, which can be utilized to control wave propagation characteristics. A number of specimens fabricated with three-dimensional printing have been tested for validation of the numerically estimated effective mechanical properties of Zongzi microstructures. The present study provides insightful information toward a new three-dimensional metastructure with tunable mechanical properties and frequency band structures.

Measuring the forces of individual muscles in a muscle group around a joint is non-trivial, and researchers have suggested using surrogates for individual muscle forces instead. Traditionally, experimentalists have shown that the force output of the skeletal muscle tissue can be correlated to the intra-muscular pressure (IMP) generated by the muscle belly. However, IMP proves difficult to measure in vivo, due to variations from sensor placement and invasiveness of the procedure. Numerical biomechanical simulations offer a tool to analyze muscle contractions, enabling new insights into the correlations among non-invasive experimentally measurable quantities, such as strains and the force output. In this work, we investigate the correlations between the muscle force output, the principal, shear and volumetric strains experienced by the muscle, as well as the pressure developed within the muscle belly as the tissue undergoes isometric contractions with varying activation profiles and magnitudes. It is observed that pressure does not correlate well with force output under higher sub-maximal and maximal activation levels, especially at locations away from the center of the muscle belly due to pressure relaxation effects. This study reveals strong correlations between force output and the strains at all locations of the belly, irrespective of the type of activation considered. This observation offers evidence for further in vivo studies using experimentally measurable principal and volumetric strains in the muscle belly as proxies for the force generation by the individual muscle and consequently enables the estimation on the contribution of various muscle groups to the total force.

This study presents a novel multi-phase computational framework that integrates physics-based modeling with machine learning algorithms for comprehensive landslide analysis from failure initiation to runout prediction. The framework employs a hydro-mechanical coupled semi-Lagrangian Reproducing Kernel Particle Method (RKPM) to model large deformation and strain localization in the seepage-induced slope failure processes. Two deformation measures, the angle change Δ θ between initially orthogonal material directions and the first principal stretch λ 1 , are extracted to characterize the deformation state at each computational node. K-means clustering in the ( Δ θ , λ 1 ) feature space is employed to identify shear-band localization regions, while Support Vector Machine (SVM) algorithm is introduced to identify the failure surfaces. The discrete interface points are subsequently regularized using the Idealized Curved Surface (ICS) method to generate a smooth three-dimensional failure surface suitable for post-failure debris flow simulation using a GPU-accelerated two-phase solver, the Modeling on Shallow flows with Efficient Simulation for Two-Phase Debris Flows (MoSES_2PDF). Validation against experimental data in the seepage-induced levee failure modeling confirmed the framework’s high accuracy in capturing slope failure depths and runout patterns, with errors in the deformed geometry remaining below 4%. The field case study further demonstrated a scalable digital twin solution capable of delivering comprehensive landslide risk assessments.

Meshfree methods, including the reproducing kernel particle method (RKPM), have been widely used within the computational mechanics community to model physical phenomena in materials undergoing large deformations or extreme topology changes. RKPM shape functions and their derivatives cannot be accurately integrated with the Gauss-quadrature methods widely employed for the finite element method (FEM) and typically require sophisticated nodal integration techniques, preventing them from easily being implemented in existing FEM software. Interpolation-based methods have been developed to address similar problems with isogeometric and immersed boundary methods, allowing these techniques to be implemented within open-source finite element software. With interpolation-based methods, background basis functions are represented as linear combinations of Lagrange polynomial foreground basis functions defined upon a boundary-conforming foreground mesh. This work extends the applications of interpolation-based methods to implement RKPM within open-source finite element software. Interpolation-based RKPM is applied to several PDEs, and error convergence rates are equivalent to classic RKPM integrated using high-order Gauss-quadrature schemes. The interpolation-based method is able to exploit the continuity of the RKPM basis to solve higher-order PDEs, demonstrated through the biharmonic problem. The method is extended to multi-material problems through Heaviside enrichment schemes, using local foreground refinement to reduce geometric integration error and achieve high-order accuracy. The computational cost of interpolation-based RKPM offers significant savings over Gauss-quadrature-based meshfree methods and the method is easily implementated within existing finite element software.

Abstract We present an open-source Python framework for the shape optimization of complex shell structures using isogeometric analysis (IGA). IGA seamlessly integrates computer-aided design (CAD) and analysis models by employing non-uniform rational B-splines (NURBS) as basis functions, enabling the natural implementation of the Kirchhoff–Love shell model due to their higher order of continuity. We leverage the recently developed FEniCS-based analysis framework, PENGoLINS, for the direct structural analysis of shell structures consisting of a collection of NURBS patches through a penalty-based formulation. This contribution introduces the open-source implementation of gradient-based shape optimization for isogeometric Kirchhoff–Love shells with a modular architecture. Complex shell structures with non-matching intersections are handled using a free-form deformation (FFD) approach and a moving intersections formulation. The symbolic differentiation and code generation capabilities in FEniCS are utilized to compute the analytical derivatives. By integrating FEniCS with OpenMDAO, we build modular components that facilitate gradient-based shape optimization of shell structures. The modular architecture in this work supports future extensions and integration with other disciplines and solvers, making it highly customizable and suitable for a wide range of applications. We validate the design-analysis-optimization workflow through several benchmark problems and demonstrate its application to aircraft wing design optimization. The framework is implemented in a Python library named GOLDFISH (Gradient-based Optimization and Large-scale Design Framework for Isogeometric SHells) and the source code will be maintained at https://github.com/hanzhao2020/GOLDFISH .

Large-scale gradient-based Multidisciplinary Design Optimization (MDO) can aid in the exploration of high-dimensional design spaces for novel air vehicle concepts, thereby leading to more efficient and economic designs. This paper builds on past works where we demonstrated large-scale physics-based MDO capabilities and applied these to NASA's lift-plus-cruise electric air taxi concept. We extend this comprehensive mid-fidelity system-level optimization problem with high(er)-fidelity subsystem-level optimizations of various aircraft systems and mission phases, with the aim of further enhancing the accuracy and scope of the aforementioned system-level optimization problem: 1) Power minimization during the transition mission phase; 2) Electrical powertrain topology optimization; 3) Cell chemistry modeling and thermo-mechanical battery pack topology optimization; 4) Shell-based coupled aero-elastic wing structure optimization. An Analytical Target Cascading-like distributed MDO architecture allows us to couple the system- and subsystem-level optimizations in order to arrive at a consistent and feasible design. We find that the system-level design is most heavily impacted by the reduced battery pack-level energy densities that stem from power peaks in the mission profile. These power peaks seem to result from the inefficient lift rotor blade designs that are needed to satisfy noise constraints. We draw conclusions based on trends in our results and give recommendations for future MDO studies of electrical air taxi vehicle concepts.

We present an open-source Python framework for the shape optimization of complex shell structures using isogeometric analysis (IGA). IGA seamlessly integrates computer-aided design (CAD) and analysis models by employing non-uniform rational B-splines (NURBS) as basis functions, enabling the natural implementation of the Kirchhoff--Love shell model due to their higher order of continuity. We leverage the recently developed FEniCS-based analysis framework, PENGoLINS, for the direct structural analysis of shell structures consisting of a collection of NURBS patches through a penalty-based formulation. This contribution introduces the open-source implementation of gradient-based shape optimization for isogeometric Kirchhoff--Love shells with a modular architecture. Complex shell structures with non-matching intersections are handled using a free-form deformation (FFD) approach and a moving intersections formulation. The symbolic differentiation and code generation capabilities in FEniCS are utilized to compute the analytical derivatives. By integrating FEniCS with OpenMDAO, we build modular components that facilitate gradient-based shape optimization of shell structures. The modular architecture in this work supports future extensions and integration with other disciplines and solvers, making it highly customizable and suitable for a wide range of applications. We validate the design-analysis-optimization workflow through several benchmark problems and demonstrate its application to aircraft wing design optimization. The framework is implemented in a Python library named GOLDFISH (Gradient-based Optimization and Large-scale Design Framework for Isogeometric SHells) and the source code will be maintained at https://github.com/hanzhao2020/GOLDFISH.

While shape optimization using isogeometric shells exhibits appealing features by integrating design geometries and analysis models, challenges arise when addressing computer-aided design (CAD) geometries comprised of multiple non-uniform rational B-splines (NURBS) patches, which are common in practice. The intractability stems from surface intersections within these CAD models. In this paper, we develop an approach for shape optimization of non-matching isogeometric shells incorporating intersection movement. Separately parametrized NURBS surfaces are modeled using Kirchhoff–Love shell theory and coupled using a penalty-based formulation. The optimization scheme allows shell patches to move without preserving relative location with other members during the shape optimization. This flexibility is achieved through an implicit state function, and analytical sensitivities are derived for the relative movement of shell patches. The introduction of differentiable intersections expands the design space and overcomes challenges associated with large mesh distortion, particularly when optimal shapes involve significant movement of patch intersections in physical space. Throughout optimization iterations, all members within the shell structures maintain the NURBS geometry representation, enabling efficient integration of analysis and design models. The optimization approach leverages the multilevel design concept by selecting a refined model for accurate analysis from a coarse design model while maintaining the same geometry. We adopt several example problems to verify the effectiveness of the proposed scheme and demonstrate its applicability to the optimization of the internal stiffeners of an aircraft wing.

Abstract An interface-modified reproducing kernel particle method (IM-RKPM) is introduced in this work to allow for a direct model construction from image pixels of heterogeneous polycrystalline Li-ion battery microstructures. The interface-modified reproducing kernel (IM-RK) approximation is constructed through scaling of a kernel function by a regularized distance function in conjunction with strategic placement of interface node locations. This leads to RK shape functions with either weak or strong discontinuities across material interfaces, suitable for modeling various interface mechanics. With the placement of a triple junction node and distance-based scaling of kernel functions, the resulting IM-RK shape function also possesses proper discontinuities at the triple junctions. This IM-RK approximation effectively remedies the well-known Gibb’s oscillation in the smooth approximation of discontinuities. Different from the conventional meshfree approaches for interface discontinuities, this IM-RK approach is done without additional degrees of freedom associated with the enrichment functions, and it is formulated with the standard procedures in the RK shape function construction. This work focuses on identifying the accuracy and convergence properties of IM-RKPM for modeling the coupled electro-chemo-mechanical system. A linear patch test is formulated and numerically tested for the electro-chemo-mechanical coupled problem with a Butler–Volmer boundary condition representing the physical conditions in Li-ion battery microstructures. This is followed by verification of the optimal rates of convergence of IM-RKPM for solving the coupled problem with higher order solutions. The image-based modeling of Li-ion battery microstructures in the numerical examples demonstrates the applicability of the proposed method to realistic Li-ion battery materials modeling.

Abstract Immersed boundary methods are high-order accurate computational tools used to model geometrically complex problems in computational mechanics. While traditional finite element methods require the construction of high-quality boundary-fitted meshes, immersed boundary methods instead embed the computational domain in a structured background grid. Interpolation-based immersed boundary methods augment existing finite element software to non-invasively implement immersed boundary capabilities through extraction. Extraction interpolates the structured background basis as a linear combination of Lagrange polynomials defined on a foreground mesh, creating an interpolated basis that can be easily integrated by existing methods. This work extends the interpolation-based immersed isogeometric method to multi-material and multi-physics problems. Beginning from level-set descriptions of domain geometries, Heaviside enrichment is implemented to accommodate discontinuities in state variable fields across material interfaces. Adaptive refinement with truncated hierarchically refined B-splines (THB-splines) is used to both improve interface geometry representations and to resolve large solution gradients near interfaces. Multi-physics problems typically involve coupled fields where each field has unique discretization requirements. This work presents a novel discretization method for coupled problems through the application of extraction, using a single foreground mesh for all fields. Numerical examples illustrate optimal convergence rates for this method in both 2D and 3D, for partial differential equations representing heat conduction, linear elasticity, and a coupled thermo-mechanical problem. The utility of this method is demonstrated through image-based analysis of a composite sample, where in addition to circumventing typical meshing difficulties, this method reduces the required degrees of freedom when compared to classical boundary-fitted finite element methods.

Conventional finite element methods are known to be tedious in adaptive refinements due to their conformal regularity requirements. Further, the enrichment functions for adaptive refinements are often not readily available in general applications. This work introduces a novel neural network-enriched Partition of Unity (NN-PU) approach for solving boundary value problems via artificial neural networks with a potential energy-based loss function minimization. The flexibility and adaptivity of the NN function space are utilized to capture complex solution patterns that the conventional Galerkin methods fail to capture. The NN enrichment is constructed by combining pre-trained feature-encoded NN blocks with an additional untrained NN block. The pre-trained NN blocks learn specific local features during the offline stage, enabling efficient enrichment of the approximation space during the online stage through the Ritz-type energy minimization . The NN enrichment is introduced under the Partition of Unity (PU) framework, ensuring convergence of the proposed method. The proposed NN-PU approximation and feature-encoded transfer learning form an adaptive approximation framework, termed the neural-refinement (n-refinement), for solving boundary value problems. Demonstrated by solving various elasticity problems , the proposed method offers accurate solutions while notably reducing the computational cost compared to the conventional adaptive refinement in the mesh-based methods.

Numerical solvers of partial differential equations (PDEs) have been widely employed for simulating physical systems. However, the computational cost remains a major bottleneck in various scientific and engineering applications, which has motivated the development of reduced-order models (ROMs). Recently, machine-learning-based ROMs have gained significant popularity and are promising for addressing some limitations of traditional ROM methods, especially for advection dominated systems. In this chapter, we focus on a particular framework known as Latent Space Dynamics Identification (LaSDI), which transforms the high-fidelity data, governed by a PDE, to simpler and low-dimensional latent-space data, governed by ordinary differential equations (ODEs). These ODEs can be learned and subsequently interpolated to make ROM predictions. Each building block of LaSDI can be easily modulated depending on the application, which makes the LaSDI framework highly flexible. In particular, we present strategies to enforce the laws of thermodynamics into LaSDI models (tLaSDI), enhance robustness in the presence of noise through the weak form (WLaSDI), select high-fidelity training data efficiently through active learning (gLaSDI, GPLaSDI), and quantify the ROM prediction uncertainty through Gaussian processes (GPLaSDI). We demonstrate the performance of different LaSDI approaches on Burgers equation, a non-linear heat conduction problem, and a plasma physics problem, showing that LaSDI algorithms can achieve relative errors of less than a few percent and up to thousands of times speed-ups.

Isogeometric analysis (IGA) has emerged as a promising approach in the field of structural optimization, benefiting from the seamless integration between the computer-aided design (CAD) geometry and the analysis model by employing non-uniform rational B-splines (NURBS) as basis functions. However, structural optimization for real-world CAD geometries consisting of multiple non-matching NURBS patches remains a challenging task. In this work, we propose a unified formulation for shape and thickness optimization of separately parametrized shell structures by adopting the free-form deformation (FFD) technique, so that continuity with respect to design variables is preserved at patch intersections during optimization. Shell patches are modeled with isogeometric Kirchhoff-Love theory and coupled using a penalty-based method in the analysis. We use Lagrange extraction to link the control points associated with the B-spline FFD block and shell patches, and we perform IGA using the same extraction matrices by taking advantage of existing finite element assembly procedures in the FEniCS partial differential equation (PDE) solution library. Moreover, we enable automated analytical derivative computation by leveraging advanced code generation in FEniCS, thereby facilitating efficient gradient-based optimization algorithms. The framework is validated using a collection of benchmark problems, demonstrating its applications to shape and thickness optimization of aircraft wings with complex shell layouts.

Immersed boundary methods are high-order accurate computational tools used to model geometrically complex problems in computational mechanics. While traditional finite element methods require the construction of high-quality boundary-fitted meshes, immersed boundary methods instead embed the computational domain in a background grid. Interpolation-based immersed boundary methods augment existing finite element software to non-invasively implement immersed boundary capabilities through extraction. Extraction interpolates the background basis as a linear combination of Lagrange polynomials defined on a foreground mesh, creating an interpolated basis that can be easily integrated by existing methods. This work extends the interpolation-based immersed boundary method to multi-material and multi-physics problems. Beginning from level-set descriptions of domain geometries, Heaviside enrichment is implemented to accommodate discontinuities in state variable fields across material interfaces. Adaptive refinement with truncated hierarchical B-splines is used to both improve interface geometry representations and resolve large solution gradients near interfaces. Multi-physics problems typically involve coupled fields where each field has unique discretization requirements. This work presents a novel discretization method for coupled problems through the application of extraction, using a single foreground mesh for all fields. Numerical examples illustrate optimal convergence rates for this method in both 2D and 3D, for heat conduction, linear elasticity, and a coupled thermo-mechanical problem. The utility of this method is demonstrated through image-based analysis of a composite sample, where in addition to circumventing typical meshing difficulties, this method reduces the required degrees of freedom compared to classical boundary-fitted finite element methods.

In deep geological repositories for high level nuclear waste with close canister spacings, bentonite buffers can experience temperatures higher than 100 °C. In this range of extreme temperatures, phenomenological constitutive laws face limitations in capturing the thermo-hydraulic behavior of the bentonite, since the pre-defined functional constitutive laws often lack generality and flexibility to capture a wide range of complex coupling phenomena as well as the effects of stress state and path dependency. In this work, a deep neural network (DNN)-based soil–water retention curve (SWRC) of bentonite is introduced and integrated into a Reproducing Kernel Particle Method (RKPM) for conducting thermo-hydraulic simulations of the bentonite buffer. The DNN-SWRC model incorporates temperature as an additional input variable, allowing it to learn the relationship between suction and degree of saturation under the general non-isothermal condition, which is difficult to represent using a phenomenological SWRC. For effective modeling of the tank-scale test, new axisymmetric Reproducing Kernel basis functions enriched with singular Dirichlet enforcement representing heater placement and an effective convective heat transfer coefficient representing thin-layer composite tank construction are developed. The proposed method is demonstrated through the modeling of a tank-scale experiment involving a cylindrical layer of MX-80 bentonite exposed to central heating.

Aircraft conceptual design is a high-dimensional optimization problem, involving up to hundreds of continuous design variables and constraints. The design of novel aircraft concepts such as electric vertical takeoff and landing (eVTOL) vehicles can benefit from the systematic exploration of the design space through the use of gradient-based optimization. In this paper, we demonstrate the application of large-scale multidisciplinary design optimization (MDO) to NASA's lift-plus-cruise electric air taxi concept, using low and mid-fidelity physics-based simulations to model the aircraft. We improve the modeling fidelity from our previous work on the same air taxi concept through refined rotor-aerodynamic and aeroacoustic models as well as in-the-loop structural analysis. In addition, we expand the scope of the analysis to include several steady design conditions, a quasi-steady transition maneuver, structural sizing conditions, as well as one-engine-inoperative (OEI) scenarios, amounting to 18 design conditions. To perform the analysis and optimization, we use a newly developed software library called the Comprehensive Aircraft high-Dimensional Design Environment (CADDEE), which facilitates the integration of all discipline models. We solve a gross weight minimization problem with over 300 design variables and over 200 constraints, spanning all modeled disciplines. Results show a decrease in gross weight of 10% after a total optimization time of about 17 hours. These results demonstrate the effectiveness of applying of large-scale MDO to the aircraft conceptual design problem.

The advent of Urban Air Mobility (UAM) has necessitated a paradigm shift in aircraft design from traditional regression methods to physics-based analysis and the use of modern computational methods. This paper explores the intricacies of UAM aircraft design, acknowledging the limitations of historical empirical equations and advocating for the use of physics-based tools in the early stages of the design process. It underscores the importance of Multidisciplinary Design, Analysis, and Optimization (MDAO) as a means to integrate physics-based tools for conceptual design, facilitating decisions on configuration and sizing. The paper presents a comprehensive survey and review of computational models across various disciplines pertinent to eVTOL/UAM/AAM concepts, aiming to guide non-discipline Subject Matter Experts (SMEs) in the application of these models within an MDAO framework. Additionally, we present ongoing work on the Verification and Validation (V&V) of a soon-to-be-released toolset developed under a NASA ULI project toward designing UAM concepts using large-scale MDAO. V&V results for low-to-mid-fidelity tools are presented, demonstrating the accuracy of the tools for UAM concept analysis and design, as validated by industry leaders and comparison with authoritative sources.

Numerical modeling of localizations is a challenging task due to the evolving rough solution in which the localization paths are not predefined. Despite decades of efforts, there is a need for innovative discretization-independent computational methods to predict the evolution of localizations. In this work, an improved version of the neural network-enhanced Reproducing Kernel Particle Method (NN-RKPM) is proposed for modeling brittle fracture. In the proposed method, a background reproducing kernel (RK) approximation defined on a coarse and uniform discretization is enriched by a neural network (NN) approximation under a Partition of Unity framework. In the NN approximation, the deep neural network automatically locates and inserts regularized discontinuities in the function space. The NN-based enrichment functions are then patched together with RK approximation functions using RK as a Partition of Unity patching function. The optimum NN parameters defining the location, orientation, and displacement distribution across location together with RK approximation coefficients are obtained via the energy-based loss function minimization. To regularize the NN-RK approximation, a constraint on the spatial gradient of the parametric coordinates is imposed in the loss function. Analysis of the convergence properties shows that the solution convergence of the proposed method is guaranteed. The NN enrichment allows the modeling of evolving cracks by a fixed coarse RK discretization without adaptive refinement for enhanced computational efficiency. The effectiveness of the proposed method is demonstrated by a series of numerical examples involving damage propagation and branching.

This work presents a multi-resolution physics-informed recurrent neural network (MR PI-RNN), for simultaneous prediction of musculoskeletal (MSK) motion and parameter identification of the MSK systems. The MSK application was selected as the model problem due to its challenging nature in mapping the high-frequency surface electromyography (sEMG) signals to the low-frequency body joint motion controlled by the MSK and muscle contraction dynamics. The proposed method utilizes the fast wavelet transform to decompose the mixed frequency input sEMG and output joint motion signals into nested multi-resolution signals. The prediction model is subsequently trained on coarser-scale input-output signals using a gated recurrent unit (GRU), and then the trained parameters are transferred to the next level of training with finer-scale signals. These training processes are repeated recursively under a transfer-learning fashion until the full-scale training (i.e., with unfiltered signals) is achieved, while satisfying the underlying dynamic equilibrium. Numerical examples on recorded subject data demonstrate the effectiveness of the proposed framework in generating a physics-informed forward-dynamics surrogate, which yields higher accuracy in motion predictions of elbow flexion-extension of an MSK system compared to the case with single-scale training. The framework is also capable of identifying muscle parameters that are physiologically consistent with the subject's kinematics data.

Abstract This work presents an approach for automating the discretization and approximation procedures in constructing digital representations of composites from micro-CT images featuring intricate microstructures. The proposed method is guided by the Support Vector Machine (SVM) classification, offering an effective approach for discretizing microstructural images. An SVM soft margin training process is introduced as a classification of heterogeneous material points, and image segmentation is accomplished by identifying support vectors through a local regularized optimization problem. In addition, an Interface-Modified Reproducing Kernel Particle Method (IM-RKPM) is proposed for appropriate approximations of weak discontinuities across material interfaces. The proposed method modifies the smooth kernel functions with a regularized Heaviside function concerning the material interfaces to alleviate Gibb's oscillations. This IM-RKPM is formulated without introducing duplicated degrees of freedom associated with the interface nodes commonly needed in the conventional treatments of weak discontinuities in the meshfree methods. Moreover, IM-RKPM can be implemented with various domain integration techniques, such as Stabilized Conforming Nodal Integration (SCNI). The extension of the proposed method to 3-dimension is straightforward, and the effectiveness of the proposed method is validated through the image-based modeling of polymer-ceramic composite microstructures.

Isogeometric analysis (IGA) has emerged as a promising approach in the field of structural optimization, benefiting from the seamless integration between the computer-aided design (CAD) geometry and the analysis model by employing non-uniform rational B-splines (NURBS) as basis functions. However, structural optimization for real-world CAD geometries consisting of multiple non-matching NURBS patches remains a challenging task. In this work, we propose a unified formulation for shape and thickness optimization of separately-parametrized shell structures by adopting the free-form deformation (FFD) technique, so that continuity with respect to design variables is preserved at patch intersections during optimization. Shell patches are modeled with isogeometric Kirchhoff--Love theory and coupled using a penalty-based method in the analysis. We use Lagrange extraction to link the control points associated with the B-spline FFD block and shell patches, and we perform IGA using the same extraction matrices by taking advantage of existing finite element assembly procedures in the FEniCS partial differential equation (PDE) solution library. Moreover, we enable automated analytical derivative computation by leveraging advanced code generation in FEniCS, thereby facilitating efficient gradient-based optimization algorithms. The framework is validated using a collection of benchmark problems, demonstrating its applications to shape and thickness optimization of aircraft wings with complex shell layouts.

Numerical modeling of localizations is a challenging task due to the evolving rough solution in which the localization paths are not predefined. Despite decades of efforts, there is a need for innovative discretization-independent computational methods to predict the evolution of localizations. In this work, an improved version of the neural network-enhanced Reproducing Kernel Particle Method (NN-RKPM) is proposed for modeling brittle fracture. In the proposed method, a background reproducing kernel (RK) approximation defined on a coarse and uniform discretization is enriched by a neural network (NN) approximation under a Partition of Unity framework. In the NN approximation, the deep neural network automatically locates and inserts regularized discontinuities in the function space. The NN-based enrichment functions are then patched together with RK approximation functions using RK as a Partition of Unity patching function. The optimum NN parameters defining the location, orientation, and displacement distribution across location together with RK approximation coefficients are obtained via the energy-based loss function minimization. To regularize the NN-RK approximation, a constraint on the spatial gradient of the parametric coordinates is imposed in the loss function. Analysis of the convergence properties shows that the solution convergence of the proposed method is guaranteed. The effectiveness of the proposed method is demonstrated by a series of numerical examples involving damage propagation and branching.

A parametric adaptive physics-informed greedy Latent Space Dynamics Identification (gLaSDI) method is proposed for accurate, efficient, and robust data-driven reduced-order modeling of high-dimensional nonlinear dynamical systems. In the proposed gLaSDI framework, an autoencoder discovers intrinsic nonlinear latent representations of high-dimensional data, while dynamics identification (DI) models capture local latent-space dynamics. An interactive training algorithm is adopted for the autoencoder and local DI models, which enables identification of simple latent-space dynamics and enhances accuracy and efficiency of data-driven reduced-order modeling. To maximize and accelerate the exploration of the parameter space for the optimal model performance, an adaptive greedy sampling algorithm integrated with a physics-informed residual-based error indicator and random-subset evaluation is introduced to search for the optimal training samples on the fly. Further, to exploit local latent-space dynamics captured by the local DI models for an improved modeling accuracy with a minimum number of local DI models in the parameter space, a k-nearest neighbor convex interpolation scheme is employed. The effectiveness of the proposed framework is demonstrated by modeling various nonlinear dynamical problems, including Burgers equations, nonlinear heat conduction, and radial advection. The proposed adaptive greedy sampling outperforms the conventional predefined uniform sampling in terms of accuracy. Compared with the high-fidelity models, gLaSDI achieves 17 to 2,658x speed-up with 1 to 5% relative errors.