Computer Methods in Applied Mechanics and Engineering, Aug 1, 1988
An arbitrary Lagrangian-Eulerian (LE) Petrov-Galerkin finite elemen{technique is developed to stu... more An arbitrary Lagrangian-Eulerian (LE) Petrov-Galerkin finite elemen{technique is developed to study nonlinear viscous fluids under large free surface wave motion. A review of the kinematics and field equations from an arbitrary refe�ence is presented and since the major challenge of the ALE description lies in the mesh rezoning algorithm, various methods, including a new mixed formulation, are developed to update the mesh and lnap the moving domain in a more rational manner. Moreover, the streamline-upwind/Petrov-Galerkin formulation is implemented to accurately describe highly convective free surface flows. The effectiveness of the algorithm is demonstrated on a tsunami problem, the dam-break problem where the Reynolds number is taken as high as 3000, and a large-amplitude sloshing problem. *The support of A. Huerta by a Catalan Fellowship and the Royal E. Cabell from Northwestern University, and the support of Wing K. Liu by the National Science Foundation are gratefully acknowledged.
Computer Methods in Applied Mechanics and Engineering, Mar 1, 2004
In this part of the work, a globally compatible C n ðXÞ triangular element hierarchy is construct... more In this part of the work, a globally compatible C n ðXÞ triangular element hierarchy is constructed in the framework of reproducing kernel element method (RKEM) for arbitrary two dimensional domains. In principle, the smoothness of the globally conforming element can be made arbitrarily high ðn P 1Þ. The triangle interpolation field can interpolate the derivatives of an unknown function up to arbitrary mth order, (I m), and it can reproduce complete kth order polynomials with k P m. This is the first interpolation hierarchical structure that has ever been constructed with both minimal degrees of freedom and higher order smoothness and continuity over discretizations of a multiple dimensional domain. The performance of the newly constructed compatible element is evaluated in solving several Kirchhoff plate problems.
Mechanical science and engineering can use machine learning. However, data sets have remained rel... more Mechanical science and engineering can use machine learning. However, data sets have remained relatively scarce; fortunately, known governing equations can supplement these data. This paper summarizes and generalizes three reduced order methods: self-consistent clustering analysis, virtual clustering analysis, and FEM-clustering analysis. These approaches have two-stage structures: unsupervised learning facilitates model complexity reduction and mechanistic equations provide predictions. These predictions define databases appropriate for training neural networks. The feed forward neural network solves forward problems, e.g., replacing constitutive laws or homogenization routines. The convolutional neural network solves inverse problems or is a classifier, e.g., extracting boundary conditions or determining if damage occurs. We will explain how these networks are applied, then provide a practical exercise: topology optimization of a structure (a) with non-linear elastic material behavior and (b) under a microstructural damage constraint. This results in microstructure-sensitive designs with computational effort only slightly more than for a conventional linear elastic analysis.
Computer Methods in Applied Mechanics and Engineering, Mar 1, 2004
In this paper and its sequels, we introduce and analyze a new class of methods, collectively call... more In this paper and its sequels, we introduce and analyze a new class of methods, collectively called the reproducing kernel element method (RKEM). The central idea in the development of the new method is to combine the strengths of both finite element methods (FEM) and meshfree methods. Two distinguished features of RKEM are: the arbitrarily high order smoothness and the interpolation property of the shape functions. These properties are desirable especially in solving Galerkin weak forms of higher order partial differential equations and in treating Dirichlet boundary conditions. So unlike the FEM, there is no need for special treatment with the RKEM in solving high order equations. Compared to meshfree methods, Dirichlet boundary conditions do not present any difficulty in using the RKEM. A rigorous error analysis and convergence study of the method are presented. The performance of the method is illustrated and assessed through some numerical examples.
The concentration and detection of molecular biomarkers remain as a challenge to develop point-of... more The concentration and detection of molecular biomarkers remain as a challenge to develop point-of-care diagnostic devices. An electric field induced concentration has been studied for such purposes but with limited success due to limited efficacy. This paper presents a computational study for investigating the molecular concentration and retention efficacy of single nanowire (SNW) and dendritic nanotip (DNT) sensors. Our computational results indicate that compared to a DNT, the SNW sensor produces higher dielectrophoretic (DEP) forces in the vicinity of the terminal end of the tip. Furthermore, the magnitude of the DEP force increases exponentially as the diameter of the SNW is decreased, resulting in a further improved retention efficacy of NPs. However, the SNW sensor's concentration efficacy was not much improved for NPs smaller than 10 nm diameter when the nanowire diameter was reduced from 500 to 50 nm. Compared to the SNW, the DNT sensor showed improved concentration efficacy due to multiple points of electric field concentrations, which retard the exponential decay of the DEP force resulting in a greater widespread region where the DEP force dominates the Brownian motion forces. When oligonucleotides are used as a target particle, the DEP force can be used to elongate oligonucleotides to further enhance the concentration and retention efficacy.
Journal of Computational and Theoretical Nanoscience, May 1, 2008
We present a review of computational methods and their applications for nanoscale mechanics of ma... more We present a review of computational methods and their applications for nanoscale mechanics of materials. Brief introductory overviews of first-principles calculations and molecular dynamics simulations are first provided, followed by various applications such as fracture mechanics and plasticity at the nanoscale. Next, recent advancement of multiscale materials modeling is reviewed, spanning from electronic scale to continuum scale. Among many multiscale methods, two newly proposed methods, the multiscale continuum theory and the virtual atom cluster model, are introduced with more details.
Soon after the discovery of carbon nanotubes, it was realized that the theoretically predicted me... more Soon after the discovery of carbon nanotubes, it was realized that the theoretically predicted mechanical properties of these interesting structures-including high strength, high stiffness, low density and structural perfection-could make them ideal for a wealth of technological applications. The experimental verification, and in some cases refutation, of these predictions, along with a number of computer simulation methods applied to their modeling, has led over the past decade to an improved but by no means complete understanding of the mechanics of carbon nanotubes. We review the theoretical predictions and discuss the experimental techniques that are most often used for the challenging tasks of visualizing and manipulating these tiny structures. We also outline the computational approaches that have been taken, including ab initio quantum mechanical simulations, classical molecular dynamics, and continuum models. The development of multiscale and multiphysics models and simulation tools naturally arises as a result of the link between basic scientific research and engineering application; while this issue is still under intensive study, we present here some of the approaches to this topic. Our concentration throughout is on the exploration of mechanical properties such as Young's modulus, bending stiffness, buckling criteria, and tensile and compressive strengths. Finally, we discuss several examples of exciting applications that take advantage of these properties, including nanoropes, filled nanotubes, nanoelectromechanical systems, nanosensors, and nanotube-reinforced polymers. This review article cites 349 references.
The ion flow in nanochannels is investigated by using nanochannels in an open configuration that ... more The ion flow in nanochannels is investigated by using nanochannels in an open configuration that allows the direct observation of fluid diffusion through an optical microscope. An “open nanochannel” is a channel with the top open to air such that fluidics can be introduced from both the entrance and the top of the channels. The experimental results showed that the diffusion length of the potassium chloride and phosphate buffer decreased with their concentration. The observed behaviors were analyzed by the contact angle variation due to the electrowetting phenomena involving the interaction between electrical double layer and counter-ions in the solution.
Computer Methods in Applied Mechanics and Engineering, Apr 1, 2008
Electric field has become one of the most widely used tools for manipulating cells, biomolecules,... more Electric field has become one of the most widely used tools for manipulating cells, biomolecules, and nanoscale particles in microfluidic devices. This paper presents the theory, modeling, and experimental works on manipulation of nano and biomaterials by using an electric field and surface tension. Three-dimensional dynamic assembly of nanowires on various microelectrodes under dielectrophoretic force is presented with discussion on capillary action and electroosmosis effects in the manipulation. The various approaches to manipulate the small scale materials are addressed both numerically and experimentally. For successful prediction and analysis on nanoscale, a hierarchical and multiscale scheme for modeling fluid transportation in nanochannels is suggested. The results show that the combined effects of electric field and capillary action induced forces are crucial for precise control over nanoscale materials.
We realize parametric resonances in a nanowire mechanical system using an oscillating electric fi... more We realize parametric resonances in a nanowire mechanical system using an oscillating electric field. Resonances at drive frequencies near 2 f 0 /n, where f 0 is the nanowire's fundamental resonance frequency, for n from 1 to 4 were observed inside a scanning electron microscope, and analyzed. Such resonances were found to originate from the amplitude-dependent electric field force acting on the nanowire and can be described by the Mathieu equation, which has known regions of instability in the parameter space.
Integrating materials and manufacturing innovation, Jul 29, 2021
Design of additively manufactured metallic parts requires computational models that can predict t... more Design of additively manufactured metallic parts requires computational models that can predict the mechanical response of parts considering the microstructural, manufacturing, and operating conditions. This article documents our response to Air Force Research Laboratory (AFRL) Additive Manufacturing Modeling Challenge 3, which asks the participants to predict the mechanical response of tensile coupons of IN625 as function of microstructure and manufacturing conditions. A representative volume element (RVE) approach was coupled with a crystal plasticity material model solved within the Fast Fourier Transformation (FFT) framework for mechanics to address the challenge. During the competition, material model calibration proved to be a challenge, prompting the introduction in this manuscript of an advanced material model identification method using proper generalized decomposition (PGD). Finally, a mechanistic reduced order method called Self-consistent Clustering Analysis (SCA) is shown as a possible alternative to the FFT method for solving these problems. Apart from presenting the response analysis, some physical interpretation and assumptions associated with the modeling are discussed.
Integrating materials and manufacturing innovation, May 11, 2021
Challenge 4 of the Air Force Research Laboratory additive manufacturing modeling challenge series... more Challenge 4 of the Air Force Research Laboratory additive manufacturing modeling challenge series asks the participants to predict the grain-average elastic strain tensors of a few specific challenge grains during tensile loading, based on experimental data and extensive characterization of an IN625 test specimen. In this article, we present our strategy and computational methods for tackling this problem. During the competition stage, a characterized microstructural image from the experiment was directly used to predict the mechanical responses of certain challenge grains with a genetic algorithm-based material model identification method. Later, in the post-competition stage, a proper generalized decomposition (PGD)-based reduced order method is introduced for improved material model calibration. This data-driven reduced order method is efficient and can be used to identify complex material model parameters in the broad field of mechanics and materials science. The results in terms of absolute error have been reported for the original prediction and re-calibrated material model. The predictions show that the overall method is capable of handling large-scale computational problems for local response identification. The re-calibrated results and speed-up show promise for using PGD for material model calibration.
The hierarchical deep-learning neural network (HiDeNN) is systematically developed through the co... more The hierarchical deep-learning neural network (HiDeNN) is systematically developed through the construction of structured deep neural networks (DNNs) in a hierarchical manner, and a special case of HiDeNN for representing Finite Element Method (or HiDeNN-FEM in short) is established. In HiDeNN-FEM, weights and biases are functions of the nodal positions, hence the training process in HiDeNN-FEM includes the optimization of the nodal coordinates. This is the spirit of r-adaptivity, and it increases both the local and global accuracy of the interpolants. By fixing the number of hidden layers and increasing the number of neurons by training the DNNs, rh-adaptivity can be achieved, which leads to further improvement of the accuracy for the solutions. The generalization of rational functions is achieved by the development of three fundamental building blocks of constructing deep hierarchical neural networks. The three building blocks are linear functions, multiplication, and inversion. With these building blocks, the class of deep learning interpolation functions are demonstrated for interpolation theories such as Lagrange polynomials, NURBS, isogeometric, reproducing kernel particle method, and others. In HiDeNN-FEM, enrichment functions through the multiplication of neurons is equivalent to the enrichment in standard finite element methods, that is, generalized, extended, and partition of unity finite element methods. Numerical examples performed by HiDeNN-FEM exhibit reduced approximation error compared with the standard FEM. Finally, an outlook for the generalized HiDeNN to high-order continuity for multiple dimensions and topology optimizations are illustrated through the hierarchy of the proposed DNNs. Keywords Neural network interpolation functions • Data-driven • r-and rh-adaptivity • Fundamental building block • Rational functions (i.e. RKPM, NURBS and IGA)
Computer Modeling in Engineering & Sciences, 2021
This paper presents the concept of reduced order machine learning finite element (FE) method. In ... more This paper presents the concept of reduced order machine learning finite element (FE) method. In particular, we propose an example of such method, the proper generalized decomposition (PGD) reduced hierarchical deeplearning neural networks (HiDeNN), called HiDeNN-PGD. We described first the HiDeNN interface seamlessly with the current commercial and open source FE codes. The proposed reduced order method can reduce significantly the degrees of freedom for machine learning and physics based modeling and is able to deal with high dimensional problems. This method is found more accurate than conventional finite element methods with a small portion of degrees of freedom. Different potential applications of the method, including topology optimization, multi-scale and multi-physics material modeling, and additive manufacturing, will be discussed in the paper.
Computer Methods in Applied Mechanics and Engineering, Aug 1, 1988
An arbitrary Lagrangian-Eulerian (LE) Petrov-Galerkin finite elemen{technique is developed to stu... more An arbitrary Lagrangian-Eulerian (LE) Petrov-Galerkin finite elemen{technique is developed to study nonlinear viscous fluids under large free surface wave motion. A review of the kinematics and field equations from an arbitrary refe�ence is presented and since the major challenge of the ALE description lies in the mesh rezoning algorithm, various methods, including a new mixed formulation, are developed to update the mesh and lnap the moving domain in a more rational manner. Moreover, the streamline-upwind/Petrov-Galerkin formulation is implemented to accurately describe highly convective free surface flows. The effectiveness of the algorithm is demonstrated on a tsunami problem, the dam-break problem where the Reynolds number is taken as high as 3000, and a large-amplitude sloshing problem. *The support of A. Huerta by a Catalan Fellowship and the Royal E. Cabell from Northwestern University, and the support of Wing K. Liu by the National Science Foundation are gratefully acknowledged.
Computer Methods in Applied Mechanics and Engineering, Mar 1, 2004
In this part of the work, a globally compatible C n ðXÞ triangular element hierarchy is construct... more In this part of the work, a globally compatible C n ðXÞ triangular element hierarchy is constructed in the framework of reproducing kernel element method (RKEM) for arbitrary two dimensional domains. In principle, the smoothness of the globally conforming element can be made arbitrarily high ðn P 1Þ. The triangle interpolation field can interpolate the derivatives of an unknown function up to arbitrary mth order, (I m), and it can reproduce complete kth order polynomials with k P m. This is the first interpolation hierarchical structure that has ever been constructed with both minimal degrees of freedom and higher order smoothness and continuity over discretizations of a multiple dimensional domain. The performance of the newly constructed compatible element is evaluated in solving several Kirchhoff plate problems.
Mechanical science and engineering can use machine learning. However, data sets have remained rel... more Mechanical science and engineering can use machine learning. However, data sets have remained relatively scarce; fortunately, known governing equations can supplement these data. This paper summarizes and generalizes three reduced order methods: self-consistent clustering analysis, virtual clustering analysis, and FEM-clustering analysis. These approaches have two-stage structures: unsupervised learning facilitates model complexity reduction and mechanistic equations provide predictions. These predictions define databases appropriate for training neural networks. The feed forward neural network solves forward problems, e.g., replacing constitutive laws or homogenization routines. The convolutional neural network solves inverse problems or is a classifier, e.g., extracting boundary conditions or determining if damage occurs. We will explain how these networks are applied, then provide a practical exercise: topology optimization of a structure (a) with non-linear elastic material behavior and (b) under a microstructural damage constraint. This results in microstructure-sensitive designs with computational effort only slightly more than for a conventional linear elastic analysis.
Computer Methods in Applied Mechanics and Engineering, Mar 1, 2004
In this paper and its sequels, we introduce and analyze a new class of methods, collectively call... more In this paper and its sequels, we introduce and analyze a new class of methods, collectively called the reproducing kernel element method (RKEM). The central idea in the development of the new method is to combine the strengths of both finite element methods (FEM) and meshfree methods. Two distinguished features of RKEM are: the arbitrarily high order smoothness and the interpolation property of the shape functions. These properties are desirable especially in solving Galerkin weak forms of higher order partial differential equations and in treating Dirichlet boundary conditions. So unlike the FEM, there is no need for special treatment with the RKEM in solving high order equations. Compared to meshfree methods, Dirichlet boundary conditions do not present any difficulty in using the RKEM. A rigorous error analysis and convergence study of the method are presented. The performance of the method is illustrated and assessed through some numerical examples.
The concentration and detection of molecular biomarkers remain as a challenge to develop point-of... more The concentration and detection of molecular biomarkers remain as a challenge to develop point-of-care diagnostic devices. An electric field induced concentration has been studied for such purposes but with limited success due to limited efficacy. This paper presents a computational study for investigating the molecular concentration and retention efficacy of single nanowire (SNW) and dendritic nanotip (DNT) sensors. Our computational results indicate that compared to a DNT, the SNW sensor produces higher dielectrophoretic (DEP) forces in the vicinity of the terminal end of the tip. Furthermore, the magnitude of the DEP force increases exponentially as the diameter of the SNW is decreased, resulting in a further improved retention efficacy of NPs. However, the SNW sensor's concentration efficacy was not much improved for NPs smaller than 10 nm diameter when the nanowire diameter was reduced from 500 to 50 nm. Compared to the SNW, the DNT sensor showed improved concentration efficacy due to multiple points of electric field concentrations, which retard the exponential decay of the DEP force resulting in a greater widespread region where the DEP force dominates the Brownian motion forces. When oligonucleotides are used as a target particle, the DEP force can be used to elongate oligonucleotides to further enhance the concentration and retention efficacy.
Journal of Computational and Theoretical Nanoscience, May 1, 2008
We present a review of computational methods and their applications for nanoscale mechanics of ma... more We present a review of computational methods and their applications for nanoscale mechanics of materials. Brief introductory overviews of first-principles calculations and molecular dynamics simulations are first provided, followed by various applications such as fracture mechanics and plasticity at the nanoscale. Next, recent advancement of multiscale materials modeling is reviewed, spanning from electronic scale to continuum scale. Among many multiscale methods, two newly proposed methods, the multiscale continuum theory and the virtual atom cluster model, are introduced with more details.
Soon after the discovery of carbon nanotubes, it was realized that the theoretically predicted me... more Soon after the discovery of carbon nanotubes, it was realized that the theoretically predicted mechanical properties of these interesting structures-including high strength, high stiffness, low density and structural perfection-could make them ideal for a wealth of technological applications. The experimental verification, and in some cases refutation, of these predictions, along with a number of computer simulation methods applied to their modeling, has led over the past decade to an improved but by no means complete understanding of the mechanics of carbon nanotubes. We review the theoretical predictions and discuss the experimental techniques that are most often used for the challenging tasks of visualizing and manipulating these tiny structures. We also outline the computational approaches that have been taken, including ab initio quantum mechanical simulations, classical molecular dynamics, and continuum models. The development of multiscale and multiphysics models and simulation tools naturally arises as a result of the link between basic scientific research and engineering application; while this issue is still under intensive study, we present here some of the approaches to this topic. Our concentration throughout is on the exploration of mechanical properties such as Young's modulus, bending stiffness, buckling criteria, and tensile and compressive strengths. Finally, we discuss several examples of exciting applications that take advantage of these properties, including nanoropes, filled nanotubes, nanoelectromechanical systems, nanosensors, and nanotube-reinforced polymers. This review article cites 349 references.
The ion flow in nanochannels is investigated by using nanochannels in an open configuration that ... more The ion flow in nanochannels is investigated by using nanochannels in an open configuration that allows the direct observation of fluid diffusion through an optical microscope. An “open nanochannel” is a channel with the top open to air such that fluidics can be introduced from both the entrance and the top of the channels. The experimental results showed that the diffusion length of the potassium chloride and phosphate buffer decreased with their concentration. The observed behaviors were analyzed by the contact angle variation due to the electrowetting phenomena involving the interaction between electrical double layer and counter-ions in the solution.
Computer Methods in Applied Mechanics and Engineering, Apr 1, 2008
Electric field has become one of the most widely used tools for manipulating cells, biomolecules,... more Electric field has become one of the most widely used tools for manipulating cells, biomolecules, and nanoscale particles in microfluidic devices. This paper presents the theory, modeling, and experimental works on manipulation of nano and biomaterials by using an electric field and surface tension. Three-dimensional dynamic assembly of nanowires on various microelectrodes under dielectrophoretic force is presented with discussion on capillary action and electroosmosis effects in the manipulation. The various approaches to manipulate the small scale materials are addressed both numerically and experimentally. For successful prediction and analysis on nanoscale, a hierarchical and multiscale scheme for modeling fluid transportation in nanochannels is suggested. The results show that the combined effects of electric field and capillary action induced forces are crucial for precise control over nanoscale materials.
We realize parametric resonances in a nanowire mechanical system using an oscillating electric fi... more We realize parametric resonances in a nanowire mechanical system using an oscillating electric field. Resonances at drive frequencies near 2 f 0 /n, where f 0 is the nanowire's fundamental resonance frequency, for n from 1 to 4 were observed inside a scanning electron microscope, and analyzed. Such resonances were found to originate from the amplitude-dependent electric field force acting on the nanowire and can be described by the Mathieu equation, which has known regions of instability in the parameter space.
Integrating materials and manufacturing innovation, Jul 29, 2021
Design of additively manufactured metallic parts requires computational models that can predict t... more Design of additively manufactured metallic parts requires computational models that can predict the mechanical response of parts considering the microstructural, manufacturing, and operating conditions. This article documents our response to Air Force Research Laboratory (AFRL) Additive Manufacturing Modeling Challenge 3, which asks the participants to predict the mechanical response of tensile coupons of IN625 as function of microstructure and manufacturing conditions. A representative volume element (RVE) approach was coupled with a crystal plasticity material model solved within the Fast Fourier Transformation (FFT) framework for mechanics to address the challenge. During the competition, material model calibration proved to be a challenge, prompting the introduction in this manuscript of an advanced material model identification method using proper generalized decomposition (PGD). Finally, a mechanistic reduced order method called Self-consistent Clustering Analysis (SCA) is shown as a possible alternative to the FFT method for solving these problems. Apart from presenting the response analysis, some physical interpretation and assumptions associated with the modeling are discussed.
Integrating materials and manufacturing innovation, May 11, 2021
Challenge 4 of the Air Force Research Laboratory additive manufacturing modeling challenge series... more Challenge 4 of the Air Force Research Laboratory additive manufacturing modeling challenge series asks the participants to predict the grain-average elastic strain tensors of a few specific challenge grains during tensile loading, based on experimental data and extensive characterization of an IN625 test specimen. In this article, we present our strategy and computational methods for tackling this problem. During the competition stage, a characterized microstructural image from the experiment was directly used to predict the mechanical responses of certain challenge grains with a genetic algorithm-based material model identification method. Later, in the post-competition stage, a proper generalized decomposition (PGD)-based reduced order method is introduced for improved material model calibration. This data-driven reduced order method is efficient and can be used to identify complex material model parameters in the broad field of mechanics and materials science. The results in terms of absolute error have been reported for the original prediction and re-calibrated material model. The predictions show that the overall method is capable of handling large-scale computational problems for local response identification. The re-calibrated results and speed-up show promise for using PGD for material model calibration.
The hierarchical deep-learning neural network (HiDeNN) is systematically developed through the co... more The hierarchical deep-learning neural network (HiDeNN) is systematically developed through the construction of structured deep neural networks (DNNs) in a hierarchical manner, and a special case of HiDeNN for representing Finite Element Method (or HiDeNN-FEM in short) is established. In HiDeNN-FEM, weights and biases are functions of the nodal positions, hence the training process in HiDeNN-FEM includes the optimization of the nodal coordinates. This is the spirit of r-adaptivity, and it increases both the local and global accuracy of the interpolants. By fixing the number of hidden layers and increasing the number of neurons by training the DNNs, rh-adaptivity can be achieved, which leads to further improvement of the accuracy for the solutions. The generalization of rational functions is achieved by the development of three fundamental building blocks of constructing deep hierarchical neural networks. The three building blocks are linear functions, multiplication, and inversion. With these building blocks, the class of deep learning interpolation functions are demonstrated for interpolation theories such as Lagrange polynomials, NURBS, isogeometric, reproducing kernel particle method, and others. In HiDeNN-FEM, enrichment functions through the multiplication of neurons is equivalent to the enrichment in standard finite element methods, that is, generalized, extended, and partition of unity finite element methods. Numerical examples performed by HiDeNN-FEM exhibit reduced approximation error compared with the standard FEM. Finally, an outlook for the generalized HiDeNN to high-order continuity for multiple dimensions and topology optimizations are illustrated through the hierarchy of the proposed DNNs. Keywords Neural network interpolation functions • Data-driven • r-and rh-adaptivity • Fundamental building block • Rational functions (i.e. RKPM, NURBS and IGA)
Computer Modeling in Engineering & Sciences, 2021
This paper presents the concept of reduced order machine learning finite element (FE) method. In ... more This paper presents the concept of reduced order machine learning finite element (FE) method. In particular, we propose an example of such method, the proper generalized decomposition (PGD) reduced hierarchical deeplearning neural networks (HiDeNN), called HiDeNN-PGD. We described first the HiDeNN interface seamlessly with the current commercial and open source FE codes. The proposed reduced order method can reduce significantly the degrees of freedom for machine learning and physics based modeling and is able to deal with high dimensional problems. This method is found more accurate than conventional finite element methods with a small portion of degrees of freedom. Different potential applications of the method, including topology optimization, multi-scale and multi-physics material modeling, and additive manufacturing, will be discussed in the paper.
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Papers by Wing Kam Liu