arXiv subjects

Wing Kam Liu

Publications and source records attributed to Wing Kam Liu.

30 records · Page 2Linked to original sources

arXiv2023

Extended tensor decomposition model reduction methods: training, prediction, and design under uncertainty

This paper introduces an extended tensor decomposition (XTD) method for model reduction. The proposed method is based on a sparse non-separated enrichment to the conventional tensor decomposition, which is expected to improve the approximation accuracy and the reducibility (compressibility) in highly nonlinear and singular cases. The proposed XTD method can be a powerful tool for solving nonlinear space-time parametric problems. The method has been successfully applied to parametric elastic-plastic problems and real time additive manufacturing residual stress predictions with uncertainty quantification. Furthermore, a combined XTD-SCA (self-consistent clustering analysis) strategy has been presented for multi-scale material modeling, which enables real time multi-scale multi-parametric simulations. The efficiency of the method is demonstrated with comparison to finite element analysis. The proposed method enables a novel framework for fast manufacturing and material design with uncertainties.

math.NA↗

arXiv2022

JAX-FEM: A differentiable GPU-accelerated 3D finite element solver for automatic inverse design and mechanistic data science

This paper introduces JAX-FEM, an open-source differentiable finite element method (FEM) library. Constructed on top of Google JAX, a rising machine learning library focusing on high-performance numerical computing, JAX-FEM is implemented with pure Python while scalable to efficiently solve problems with moderate to large sizes. For example, in a 3D tensile loading problem with 7.7 million degrees of freedom, JAX-FEM with GPU achieves around 10$\times$ acceleration compared to a commercial FEM code depending on platform. Beyond efficiently solving forward problems, JAX-FEM employs the automatic differentiation technique so that inverse problems are solved in a fully automatic manner without the need to manually derive sensitivities. Examples of 3D topology optimization of nonlinear materials are shown to achieve optimal compliance. Finally, JAX-FEM is an integrated platform for machine learning-aided computational mechanics. We show an example of data-driven multi-scale computations of a composite material where JAX-FEM provides an all-in-one solution from microscopic data generation and model training to macroscopic FE computations. The source code of the library and these examples are shared with the community to facilitate computational mechanics research.

cs.MS↗

arXiv2022

An Introduction to Kernel and Operator Learning Methods for Homogenization by Self-consistent Clustering Analysis

Recent advances in operator learning theory have improved our knowledge about learning maps between infinite dimensional spaces. However, for large-scale engineering problems such as concurrent multiscale simulation for mechanical properties, the training cost for the current operator learning methods is very high. The article presents a thorough analysis on the mathematical underpinnings of the operator learning paradigm and proposes a kernel learning method that maps between function spaces. We first provide a survey of modern kernel and operator learning theory, as well as discuss recent results and open problems. From there, the article presents an algorithm to how we can analytically approximate the piecewise constant functions on R for operator learning. This implies the potential feasibility of success of neural operators on clustered functions. Finally, a k-means clustered domain on the basis of a mechanistic response is considered and the Lippmann-Schwinger equation for micro-mechanical homogenization is solved. The article briefly discusses the mathematics of previous kernel learning methods and some preliminary results with those methods. The proposed kernel operator learning method uses graph kernel networks to come up with a mechanistic reduced order method for multiscale homogenization.

cs.LG↗

arXiv2022

Deep Learning Discrete Calculus (DLDC): A Family of Discrete Numerical Methods by Universal Approximation for STEM Education to Frontier Research

The article proposes formulating and codifying a set of applied numerical methods, coined as Deep Learning Discrete Calculus (DLDC), that uses the knowledge from discrete numerical methods to interpret the deep learning algorithms through the lens of applied mathematics. The DLDC methods aim to leverage the flexibility and ever increasing resources of deep learning and rich literature of numerical analysis to formulate a general class of numerical method that can directly use data with uncertainty to predict the behavior of an unknown system as well as elevate the speed and accuracy of numerical solution of the governing equations for known systems. The article is structured in two major sections. In the first section, the building blocks of the DLDC methods are presented and deep learning structures analogous to traditional numerical methods such as finite difference and finite element methods are constructed with a view to incorporate these techniques in Science, Technology, Engineering, Mathematics (STEM) syllabus for K-12 students. The second section builds upon the building blocks of the previous discussion,and proposes new solution schemes for differential and integral equations pertinent to multiscale mechanics. Each section is accompanied with mathematical formulation of the numerical methods, analogous DLDC formulation, and suitable examples.

math.NA↗

arXiv2022

Multiscale reduced-order modeling of fused filament fabricated composites

Defects such as voids are observed at multiple length scales of an additively manufactured composite material. Modeling such defects and their multiscale interaction is crucial for the materials performance prediction. In this work, we study as-built defects in fused filament fabricated Polycarbonate/Short Carbon Fiber (PC/SCF) composite samples. The microscale and mesoscale voids along with the mesoscale layer orientations have been studied using a mechanistic reduced-order model. Our result indicates that the microscale intrabead voids interact with the mesoscale interbead voids and significantly degrade the mechanical response of the printed composites compared to the microscale microstructure without voids. The mesoscale layer orientations also influence the stress-strain response and show better performance when the load is applied to the bead direction. The efficient reduced-order modeling approach used in this work provides a way to evaluate multiscale design aspects of additively manufactured composite materials.

cond-mat.mtrl-sci↗

arXiv2022

Stochastic additive manufacturing simulation: from experiment to surface roughness and porosity prediction

Deterministic computational modeling of laser powder bed fusion (LPBF) process fails to capture irregularities and roughness of the scan track, unless expensive powder-scale analysis is used. In this work we developed a stochastic computational modeling framework based on Markov Chain Monte Carlo (MCMC) capable of capturing the irregularities of LPBF scan. The model is calibrated against AFRL single track scan data using a specially designed tensor decomposition method, i.e., Higher-Order Proper Generalized Decomposition (HOPGD) that relies on non-intrusive data learning and construction of reduced order surrogate models. Once calibrated, the stochastic model can be used to predict the roughness and porosity at part scale at a significantly reduced computational cost compared to detailed powder-scale deterministic simulations. The stochastic simulation predictions are validated against AFRL multi-layer and multitrack experiments and reported as more accurate when compared with regular deterministic simulation results.

cs.CE↗

arXiv2021

Concurrent n-scale modeling for non-orthogonal woven composite

Concurrent analysis of composite materials can provide the interaction among scales for better composite design, analysis, and performance prediction. A data-driven concurrent n-scale modeling theory ($\textrm{FExSCA}^\textrm{n-1}$) is proposed in this paper utilizing a mechanistic reduced order model (ROM) called self-consistent clustering analysis (SCA). We demonstrated this theory with a $\textrm{FExSCA}^2$ approach to study the 3-scale woven carbon fiber reinforced polymer (CFRP) laminate structure. $\textrm{FExSCA}^2$ significantly reduced expensive 3D nested composite representative volume elements (RVEs) computation for woven and unidirectional (UD) composite structures by developing a material database. The modeling procedure is established by integrating the material database into a woven CFRP structural numerical model, formulating a concurrent 3-scale modeling framework. This framework provides an accurate prediction for the structural performance (e.g., nonlinear structural behavior under tensile load), as well as the woven and UD physics field evolution. The concurrent modeling results are validated against physical tests that link structural performance to the basic material microstructures. The proposed methodology provides a comprehensive predictive modeling procedure applicable to general composite materials aiming to reduce laborious experiments needed.

cond-mat.mtrl-sci↗

arXiv2021

Mechanistic Data Science for Modeling and Design of Aerospace Composite Materials

Polymer matrix composites exhibit remarkable lightweight and high strength properties that make them attractive for aerospace applications. Constituents' materials such as advanced polymers and fibers or fillers with their hierarchical structure embed these exceptional properties to the composite materials. This hierarchical structure in multiple length scales provides an opportunity for designing the composite materials for optimized properties. However, the high dimensional design space for the constituents' materials and architectures choice of the composites makes it a challenging design problem. To tackle this high dimensional design space, a systematic, efficient approach named mechanistic data science framework is proposed in this work to identify the governing mechanisms of materials systems from the limited available data and create a composite knowledge database. Our composite knowledge database comprises the knowledge of polymers at the nanoscale with nano reinforcement, the unidirectional structure at the microscale, and woven structure at mesoscale mechanisms that can be further used for the part scale composite design and analysis. The mechanistic data science framework presented in this work can be further extended to other materials systems that will provide the materials designer with the necessary tools to evaluate a new materials system design rapidly.

cond-mat.mtrl-sci↗

arXiv2021

Data-driven discovery of dimensionless numbers and scaling laws from experimental measurements

Dimensionless numbers and scaling laws provide elegant insights into the characteristic properties of physical systems. Classical dimensional analysis and similitude theory fail to identify a set of unique dimensionless numbers for a highly-multivariable system with incomplete governing equations. In this study, we embed the principle of dimensional invariance into a two-level machine learning scheme to automatically discover dominant and unique dimensionless numbers and scaling laws from data. The proposed methodology, called dimensionless learning, can reduce high-dimensional parametric spaces into descriptions involving just a few physically-interpretable dimensionless parameters, which significantly simplifies the process design and optimization of the system. We demonstrate the algorithm by solving several challenging engineering problems with noisy experimental measurements (not synthetic data) collected from the literature. The examples include turbulent Rayleigh-Benard convection, vapor depression dynamics in laser melting of metals, and porosity formation in 3D printing. We also show that the proposed approach can identify dimensionally-homogeneous differential equations with minimal parameters by leveraging sparsity-promoting techniques.

physics.flu-dyn↗

arXiv2021

Eighty Years of the Finite Element Method: Birth, Evolution, and Future

This year marks the eightieth anniversary of the invention of the finite element method (FEM). FEM has become the computational workhorse for engineering design analysis and scientific modeling of a wide range of physical processes, including material and structural mechanics, fluid flow and heat conduction, various biological processes for medical diagnosis and surgery planning, electromagnetics and semi-conductor circuit and chip design and analysis, additive manufacturing, i.e. virtually every conceivable problem that can be described by partial differential equations (PDEs). FEM has fundamentally revolutionized the way we do scientific modeling and engineering design, ranging from automobiles, aircraft, marine structures, bridges, highways, and high-rise buildings. Associated with the development of finite element methods has been the concurrent development of an engineering science discipline called computational mechanics, or computational science and engineering. In this paper, we present a historical perspective on the developments of finite element methods mainly focusing on its applications and related developments in solid and structural mechanics, with limited discussions to other fields in which it has made significant impact, such as fluid mechanics, heat transfer, and fluid-structure interaction. To have a complete storyline, we divide the development of the finite element method into four time periods: I. (1941-1965) Early years of FEM; II. (1966-1991) Golden age of FEM; III. (1992-2017) Large scale, industrial applications of FEM and development of material modeling, and IV (2018-) the state-of-the-art FEM technology for the current and future eras of FEM research. Note that this paper may not strictly follow the chronological order of FEM developments, because often time these developments were interwoven across different time periods.

math.NA↗

arXiv2021

HiDeNN-PGD: reduced-order hierarchical deep learning neural networks

This paper presents a proper generalized decomposition (PGD) based reduced-order model of hierarchical deep-learning neural networks (HiDeNN). The proposed HiDeNN-PGD method keeps both advantages of HiDeNN and PGD methods. The automatic mesh adaptivity makes the HiDeNN-PGD more accurate than the finite element method (FEM) and conventional PGD, using a fraction of the FEM degrees of freedom. The accuracy and convergence of the method have been studied theoretically and numerically, with a comparison to different methods, including FEM, PGD, HiDeNN and Deep Neural Networks. In addition, we theoretically showed that the PGD converges to FEM at increasing modes, and the PGD error is a direct sum of the FEM error and the mode reduction error. The proposed HiDeNN-PGD performs high accuracy with orders of magnitude fewer degrees of freedom, which shows a high potential to achieve fast computations with a high level of accuracy for large-size engineering problems.

math.NA↗

arXiv2016

Ternary and senary representations using DNA double-crossover tiles

The information capacity of double-crossover (DX) tiles was successfully increased beyond a binary representation to higher base representations. By controlling the length and the position of DNA hairpins on the DX tile, ternary and senary (base-3 and base-6) digit representations were realized and verified by atomic force microscopy (AFM). Also, normal mode analysis (NMA) was carried out to study the mechanical characteristics of each structure.

physics.bio-ph↗