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Direct representation of material microstructure in a macroscale simulation is prohibitively expensive, if even possible, with current methods. However, the information contained in such a representation is highly desirable for tasks such as material/alloy design and manufacturing process control. In this paper, a mechanistic machine learning framework is developed for fast multiscale analysis of material response and structure performance. The new capabilities stem from three major factors: (1) the use of an unsupervised learning (clustering)-based discretization to achieve significant order reduction at both macroscale and microscale; (2) the generation of a database of interaction tensors among discretized material regions; (3) concurrent multiscale response prediction to solve the mechanistic equations. These factors allow for an orders-of-magnitude decrease in the computational expense compared to FEn, n $$\ge $$ 2. This method provides sufficiently high fidelity and speed to reasonably conduct inverse modeling for the challenging tasks mentioned above.

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.

In this paper, we derive an analytical expression for reproducing kernel particle method (RKPM) shape functions. Based on this, we propose a necessary and sufficient stability condition for general RKPM in arbitrary function space, and illustrate with degenerate cases. By selecting proper basis vectors and the support of the kernel functions, we demonstrate that the RKPM framework allows generating many kinds of shape functions, including the Lagrangian, B-spline and NURBS shape functions.

Accurate and efficient modeling of microstructural interaction and evolution for prediction of the macroscopic behavior of materials is important for material design and manufacturing process control . This paper approaches this challenge with a reduced-order method called self-consistent clustering analysis (SCA). It is reformulated for general elasto-viscoplastic materials under large deformation . The accuracy and efficiency for predicting overall mechanical response of polycrystalline materials is demonstrated with a comparison to traditional full-field solution methods such as finite element analysis and the fast Fourier transform . It is shown that the reduced-order method enables fast prediction of microstructure–property relationships with quantified variation. The utility of the method is demonstrated by conducting a concurrent multiscale simulation of a large-deformation manufacturing process with sub-grain spatial resolution while maintaining reasonable computational expense. This method could be used for microstructure-sensitive properties design as well as process parameters optimization.

Recently proposed clustering-based methods considerably reduce numerical cost for homogenizing heterogeneous materials, while maintaining the accuracy of mechanical property predictions in an online stage. In such an algorithm, however, the calculation of interaction tensors consumes much of the total computing time. We introduce a new method that expedites the interaction tensors calculation, thereby enhancing the clustering-based methods. We first cast a cubic/rectangular coarse grid over the representative volume element. Using analytical expressions for the integral of the Green’s functions, we then calculate interaction tensors on the coarse grid. Finally, the desired interaction tensors on the clusters are approximated based on composition ratios. Moreover, in virtual clustering analysis, we derive the Lippmann–Schwinger equation for finite strain problems. Numerical tests in two and three space dimensions verify the efficiency and accuracy of the proposed method.

A new data-driven method that generalizes experimentally measured and/or computational generated data sets under different loading paths to build three dimensional nonlinear elastic material law with objectivity under arbitrary loadings using neural networks is proposed. The proposed approach is first demonstrated by exploiting the concept of representative volume element (RVE) in the principal strain and stress spaces to numerically generate the data. A computational data-training algorithm on the generalization of these principal space data to three dimensional objective isotropic material laws subjected to arbitrary deformation is given. To validate these data-driven derived material laws, large deformation and buckling analysis of nonlinear elastic solids with reference material models and engineering structure with microstructure are performed. Numerical experiments show that only seven sets of data under different stress loading paths on RVEs are required to reach reasonable accuracy. The requirements for constitutive law such as objectivity are preserved approximately. The consistent tangent modulus is also derived. The proposed approach also shows a great potential to obtain the material law between different scales in the multiscale analysis by pure data.

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.

In order to represent the functionally graded properties of interphase, a multi-coated micromechanical model is developed. Based on elliptic shell integration of Green’s function, the strain disturbance in each phase is obtained. According to computational investigation of this model, the outer layer of the interphase does not bring in strain disturbance within the inner ones. To this end, a sequential computational homogenization method is proposed. The inhomogeneities are added sequentially from outside to inside. The temporary effective modulus on each stage is obtained by the Self Consistency Scheme. Then the effective modulus of the overall composites are fitted with a Mori–Tanaka estimation for practical applications. The effectiveness of present method is verified by the results of “2 + 1” and “3 + 1” models in prior researches and finite element simulations. Finally, the influence of thickness and stiffness of interphase on the composites’ effective modulus are investigated.

Phase-field models for fracture problems have attracted considerable attention in recent years, which are capable of tracking the discontinuities numerically, and also produce complex crack patterns in many applications. In this paper, a phase-field model for a general nonlinearly elastic material is proposed using a novel additive decomposition of strain energy. This decomposition has two parts: one is principal stretch related and the other solely composed of volumetric deformation, which accounts for different behaviors of fracture in tension and compression. We construct the Lagrangian by integrating the split energies and the separation energy from phase-field approximation for discrete cracks. A coupled system of equations is also derived that governs the deformation of the body and the evolution of phase field. The capability and performance of the proposed model are demonstrated in several representative examples. Our results show that the predicted fracture surfaces are in good agreement with experimental observations. Compared with the previous models in which the energy is simply split into the isochoric and volumetric parts, the present model is numerically more robust and effective in simulating sharp cracks. The present model can also aid researchers to control the degree of tension–compression asymmetry in the nonlinear regime of deformation, which can be naturally extended to simulate the fracture of the rubber-like materials with tension–compression asymmetry.

Scoliosis, an abnormal curvature of the human spinal column, is characterized by a lateral deviation of the spine, accompanied by axial rotation of the vertebrae. Adolescent Idiopathic Scoliosis (AIS) is the most common type, affecting children between ages 8 to 18 when bone growth is at its maximum rate. We propose a mechanistic machine learning algorithm in order to study patient-specific AIS curve progression, which is associated with the bone growth and other genetic and environmental factors. Two different frameworks are used to analyse and predict curve progression, one with implementing clinical data extracted from 2D X-ray images and the other one with incorporating both clinical data and physical equations governing the non-uniform bone growth. The physical equations governing bone growth are affiliated with calculating all stress components at each region. The stress values are evaluated through a surrogate finite element simulation and a bone growth model on a detailed patient-specific geometry of the human spine. We also propose a patient-specific framework to generate the volumetric model of human spine which is partitioned into different tissues for both vertebra and intervertebral disc. It is shown that implementing physical equations governing bone growth into the prediction framework will notably improve the prediction results as compared to only using clinical data for prediction. In addition, we can predict curve progression at ages outside the range of training samples.

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., Comment: 35 pages, 12 figures

An arbitrary Lagrangian-Eulerian (ALE) Petrov-Galerkin finite element 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 reference 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 map 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.

International audience; A mechanical science of materials, based on data science, is formulated to predict process-structure-property-performance relationships. Sampling techniques are used to build a training database, which is then compressed using unsupervised learning methods, and finally used to generate predictions by means of supervised learning methods or mechanistic equations. The method presented in this paper relies on an a priori deterministic sampling of the solution space, a K-means clustering method, and a mechanistic Lippmann-Schwinger equation solved using a self-consistent scheme. This method is formulated in a finite strain setting in order to model the large plastic strains that develop during metal forming processes. An efficient implementation of an inclusion fragmentation model is introduced in order to model this micromechanism in a clustered discretization. With the addition of a fatigue strength prediction method also based on data science, process-structure-property-performance relationships can be predicted in the case of cold-drawn NiTi tubes.

One goal of modeling for metal Additive Manufacturing (AM) is to predict the resultant mechanical properties from given manufacturing process parameters and intrinsic material properties, thereby reducing uncertainty in the material built. This can dramatically reduce the time and cost for the development of new products using AM. We have realized the seamless linking of models for the manufacturing process, material structure formation, and mechanical response through an integrated multi-physics modeling framework. The sequentially coupled modeling framework relies on the concept that the results from each model used in the framework are contained in space-filling volume elements using a prescribed structure. This framework is implemented to show a prediction of the decrease in fatigue life caused by insufficient fusion resulting from low laser power relative to the hatch spacing. In this demonstration, powder spreading and thermal-fluid flow models are used to predict the thermal history and void formation in a multilayer, multi-track build with different processing conditions. The results of these predictions are passed to a cellular automaton-based prediction of grain structure. Finally, the predicted grain and void structure is passed to a reduced-order micromechanics-based model to predict mechanical properties and fatigue life arising from the different processing conditions used in the process model. The simulation results from this combination of models demonstrate qualitative agreement with experimental observations from literature, showing the appealing potential of an integrated framework.

As a promising powder-based additive manufacturing technology, selective laser melting (SLM) has gained great popularity in recent years. However, experimental observation of the melting and solidification process is very challenging. This hinders the study of the physical mechanisms behind a variety of phenomena in SLM such as splashing and balling effects, and further poses challenges to the quality control of the products. Powder-scale computational models can reproduce the multi-physics process of SLM. In this study, we couple the Finite Volume Method (FVM) and Discrete Element Method to model the deposition of powder particles, and use the FVM to model the melting process, both with ambient air. In particular, a cutting-edge sharp surface capturing technique (iso-Advector) is incorporated into the Volume of Fluid Model to reconstruct the interface between different phases during the melting process. Iso-Advector is then used to capture and reconstruct the interface between molten material and ambient air, which is further used as a solid boundary for spreading the next powder layer. As such, 3D geometrical data is exchanged between these two stages repeatedly to reproduce the powder spreading-melting process of SLM incorporating different scan paths on multiple powder layers. To demonstrate the effectiveness of the powder-scale multi-physics modeling framework, typical scenarios with different fabrication parameters (Ti–6Al–4V powder) are simulated and compared with experimental observations available in literature.

The multiresolution continuum theory (MCT) is implemented in FEA with a bespoke user defined element and materials. A simple dog-bone model is used to validate the code and study the effect of microscale parameters. The ability of the method to simulate the propagation of a shear band in simple shear without mesh dependence is shown. The length scale parameter is demonstrated to influence shear band width. Finally, we present a simulation of serrated chip formation in metal cutting, a case where accurate prediction of shear band formation is critical. The advantages of MCT over conventional methods are discussed. This work helps elucidate the role of the length scale and microscale parameters in MCT, and is a demonstration of a practical engineering application of the method: the simulation of high speed cutting.

Woven fiber composites have been increasingly employed as light-weight materials in aerospace, construction, and transportation industries due to their superior properties. These materials possess a hierarchical structure that necessitates the use of multiscale simulations in their modeling. To account for the inherent uncertainty in materials, such simulations must be integrated with statistical uncertainty quantification (UQ) and propagation (UP) methods. However, limited advancement has been made in this regard due to the significant computational costs and complexities in modeling spatially correlated structural variations coupled at different scales. In this work, a non-intrusive approach is proposed for multiscale UQ and UP to address these limitations. We introduce the top-down sampling method that allows to model non-stationary and continuous (but not differentiable) spatial variations of uncertainty sources by creating nested random fields (RFs) where the hyperparameters of an ensemble of RFs is characterized by yet another RF. We employ multi-response Gaussian RFs in top-down sampling and leverage statistical techniques (such as metamodeling and dimensionality reduction) to address the considerable computational costs of multiscale simulations. We apply our approach to quantify the uncertainty in a cured woven composite due to spatial variations of yarn angle, fiber volume fraction, and fiber misalignment angle. Our results indicate that, even in linear analysis, the effect of uncertainty sources on the material’s response could be significant.

In this paper, we propose a new homogenization algorithm, virtual clustering analysis (VCA), as well as provide a mathematical framework for the recently proposed self-consistent clustering analysis (SCA) (Liu et al. in Comput Methods Appl Mech Eng 306:319–341, 2016). In the mathematical theory, we clarify the key assumptions and ideas of VCA and SCA, and derive the continuous and discrete Lippmann–Schwinger equations. Based on a key postulation of “once response similarly, always response similarly”, clustering is performed in an offline stage by machine learning techniques (k-means and SOM), and facilitates substantial reduction of computational complexity in an online predictive stage. The clear mathematical setup allows for the first time a convergence study of clustering refinement in one space dimension. Convergence is proved rigorously, and found to be of second order from numerical investigations. Furthermore, we propose to suitably enlarge the domain in VCA, such that the boundary terms may be neglected in the Lippmann–Schwinger equation, by virtue of the Saint-Venant’s principle. In contrast, they were not obtained in the original SCA paper, and we discover these terms may well be responsible for the numerical dependency on the choice of reference material property. Since VCA enhances the accuracy by overcoming the modeling error, and reduce the numerical cost by avoiding an outer loop iteration for attaining the material property consistency in SCA, its efficiency is expected even higher than the recently proposed SCA algorithm.

Multiscale modeling of heterogeneous material undergoing strain softening poses computational challenges for localization of the microstructure, material instability in the macrostructure, and the computational requirement for accurate and efficient concurrent calculation. In the paper, a stable micro-damage homogenization algorithm is presented which removes the material instability issues in the microstructure with representative volume elements (RVE) that are not sensitive to size when computing the homogenized stress–strain response. The proposed concurrent simulation framework allows the computation of the macroscopic response to explicitly consider the behavior of the separate constituents (material phases), as well as the complex microstructural morphology. A non-local material length parameter is introduced in the macroscale model, which will control the width of the damage bands and prevent material instability. The self-consistent clustering analysis (SCA) recently proposed by Liu et al. [ 1 ] provides an effective way of developing a microstructural database based on a clustering algorithm and the Lippmann–Schwinger integral equation, which enables an efficient and accurate prediction of nonlinear material response. The self-consistent clustering analysis is further generalized to consider complex loading paths through the projection of the effective stiffness tensor. In the concurrent simulation, the predicted macroscale strain localization is observed to be sensitive to the combination of microscale constituents, showing the unique capability of the SCA microstructural database for complex material simulations.

The reproducing kernel particle method (RKPM) has been efficiently applied to problems with large deformations, high gradients and high modal density. In this paper, it is extended to solve a nonlocal problem modeled by a fractional advection–diffusion equation (FADE), which exhibits a boundary layer with low regularity. We formulate this method on a moving least-square approach. Via the enrichment of fractional-order power functions to the traditional integer-order basis for RKPM, leading terms of the solution to the FADE can be exactly reproduced, which guarantees a good approximation to the boundary layer. Numerical tests are performed to verify the proposed approach.

In this paper, a parallelized 3D cellular automaton computational model is developed to predict grain morphology for solidification of metal during the additive manufacturing process. Solidification phenomena are characterized by highly localized events, such as the nucleation and growth of multiple grains. As a result, parallelization requires careful treatment of load balancing between processors as well as interprocess communication in order to maintain a high parallel efficiency. We give a detailed summary of the formulation of the model, as well as a description of the communication strategies implemented to ensure parallel efficiency. Scaling tests on a representative problem with about half a billion cells demonstrate parallel efficiency of more than 80% on 8 processors and around 50% on 64; loss of efficiency is attributable to load imbalance due to near-surface grain nucleation in this test problem. The model is further demonstrated through an additive manufacturing simulation with resulting grain structures showing reasonable agreement with those observed in experiments.

Additive manufacturing (AM) possesses appealing potential for manipulating material compositions, structures and properties in end-use products with arbitrary shapes without the need for specialized tooling. Since the physical process is difficult to experimentally measure, numerical modeling is a powerful tool to understand the underlying physical mechanisms. This paper presents our latest work in this regard based on comprehensive material modeling of process---structure---property relationships for AM materials. The numerous influencing factors that emerge from the AM process motivate the need for novel rapid design and optimization approaches. For this, we propose data-mining as an effective solution. Such methods--used in the process---structure, structure---properties and the design phase that connects them--would allow for a design loop for AM processing and materials. We hope this article will provide a road map to enable AM fundamental understanding for the monitoring and advanced diagnostics of AM processing.

A new data-driven computational framework is developed to assist in the design and modeling of new material systems and structures. The proposed framework integrates three general steps: (1) design of experiments, where the input variables describing material geometry (microstructure), phase properties and external conditions are sampled; (2) efficient computational analyses of each design sample, leading to the creation of a material response database; and (3) machine learning applied to this database to obtain a new design or response model. In addition, the authors address the longstanding challenge of developing a data-driven approach applicable to problems that involve unacceptable computational expense when solved by standard analysis methods – e.g. finite element analysis of representative volume elements involving plasticity and damage. In these cases the framework includes the recently developed “self-consistent clustering analysis” method in order to build large databases suitable for machine learning. The authors believe that this will open new avenues to finding innovative materials with new capabilities in an era of high-throughput computing (“big-data”).

In this paper, an enriched finite element method with fractional basis $$\left[ 1,x^{\alpha }\right] $$ for spatial fractional partial differential equations is proposed to obtain more stable and accurate numerical solutions. For pure fractional diffusion equation without advection, the enriched Galerkin finite element method formulation is demonstrated to simulate the exact solution successfully without any numerical oscillation, which is advantageous compared to the traditional Galerkin finite element method with integer basis $$\left[ 1,x\right] $$ . For fractional advection–diffusion equation, the oscillatory behavior becomes complex due to the introduction of the advection term which can be characterized by a fractional element Peclet number. For the purpose of addressing the more complex numerical oscillation, an enriched Petrov–Galerkin finite element method is developed by using a dimensionless fractional stabilization parameter, which is formulated through a minimization of the residual of the nodal solution. The effectiveness and accuracy of the enriched finite element method are demonstrated by a series of numerical examples of fractional diffusion equation and fractional advection–diffusion equation, including both one-dimensional and two-dimensional, steady-state and time-dependent cases.

Polymer nanocomposites have been envisioned as advanced materials for improving the mechanical performance of neat polymers used in aerospace, petrochemical, environment and energy industries. With the filler size approaching the nanoscale, composite materials tend to demonstrate remarkable thermomechanical properties, even with addition of a small amount of fillers. These observations confront the classical composite theories and are usually attributed to the high surface-area-to-volume-ratio of the fillers, which can introduce strong nanoscale interfacial effect and relevant long-range perturbation on polymer chain dynamics. Despite decades of research aimed at understanding interfacial effect and improving the mechanical performance of composite materials, it is not currently possible to accurately predict the mechanical properties of polymer nanocomposites directly from their molecular constituents. To overcome this challenge, different theoretical, experimental and computational schemes will be used to uncover the key physical mechanisms at the relevant spatial and temporal scales for predicting and tuning constitutive behaviors in silico, thereby establishing a bottom-up virtual design principle to achieve unprecedented mechanical performance of nanocomposites. A modular-based multiscale modeling approach for viscoelasticity of polymer nanocomposites has been proposed and discussed in this study, including four modules: (A) neat polymer toolbox; (B) interphase toolbox; (C) microstructural toolbox and (D) homogenization toolbox. Integrating these modules together, macroscopic viscoelasticity of polymer nanocomposites could be directly predicted from their molecular constituents. This will maximize the computational ability to design novel polymer composites with advanced performance. More importantly, elucidating the viscoelasticity of polymer nanocomposites through fundamental studies is a critical step to generate an integrated computational material engineering principle for discovering and manufacturing new composites with transformative impact on aerospace, automobile, petrochemical industries.

This paper presents an in-depth numerical analysis of spatial fractional advection–diffusion equations (FADE) utilizing the finite element method (FEM). A traditional Galerkin finite element formulation of the pure fractional diffusion equation without advection may yield numerical oscillations in the solution depending on the fractional derivative order. These oscillations are similar to those that may arise in the integer-order advection–diffusion equation when using the Galerkin FEM. In a Galerkin formulation of a FADE, these oscillations are further compounded by the presence of the advection term, which we show can be characterized by a fractional element Peclet number that takes into account the fractional order of the diffusion term. To address this oscillatory behavior, a Petrov–Galerkin method is formulated using a fractional stabilization parameter to eliminate the oscillatory behavior arising from both the fractional diffusion and advection terms. A compact formula for an optimal fractional stabilization parameter is developed through a minimization of the residual of the nodal solution. Steady state and transient one-dimensional cases of the pure fractional diffusion and fractional advection–diffusion equations are implemented to demonstrate the effectiveness and accuracy of the proposed formulation.

Fractional derivatives play a very important role in modeling physical phenomena involving long-range correlation effects. However, they raise challenges of computational cost and memory storage requirements when solved using current well developed numerical methods. In this paper, the differential operator multiplication method is proposed to address the issues by considering a reaction---advection---diffusion equation with a fractional derivative in time. The linear fractional differential equation is transformed into an integer order differential equation by the proposed method, which can fundamentally fix the aforementioned issues for select fractional differential equations. In such a transform, special attention should be paid to the initial conditions for the resulting differential equation of higher integer order. Through numerical experiments, we verify the proposed method for both fractional ordinary differential equations and partial differential equations.

The discovery of efficient and accurate descriptions for the macroscopic behavior of materials with complex microstructure is an outstanding challenge in mechanics of materials. A mechanistic, data-driven, two-scale approach is developed for predicting the behavior of general heterogeneous materials under irreversible processes such as inelastic deformation. The proposed approach includes two major innovations: (1) the use of a data compression algorithm, k -means clustering, during the offline stage of the method to homogenize the local features of the material microstructure into a group of clusters; and (2) a new method called self-consistent clustering analysis used in the online stage that is valid for any local plasticity laws of each material phase without the need for additional calibration. A particularly important feature of the proposed approach is that the offline stage only uses the linear elastic properties of each material phase, making it efficient. This work is believed to open new avenues in parameter-free multi-scale modeling of complex materials, and perhaps in other fields that require homogenization of irreversible processes.

This work develops a model for the mechanical response of trabecular bone including plasticity, damage and fracture. It features a resultant lamellar orientation that captures trabecular strut anisotropic elasticity, and introduces asymmetric J2 plasticity with isotropic hardening to capture evolving strut tensile and compressive dissipative properties. A continuum compatibility based damage and fracture criterion is also proposed to model fracture surface generation. We investigated fracture of a trabecular bone network under a compressive load, for which failure modes of both tension and compression were identified at the strut level. The predicted trabecular network response was found to fall within the range of experimental results reported in literature. We also investigated the response of idealized struts under compression, tension and bending using our model. Individual struts were found to exhibit micro-buckling under compression and micro-necking under tension. These instabilities are however masked by the multiplicity and complexity of strut orientations at the trabecular network level.

Direct numerical simulation based on experimental stress–strain data without explicit constitutive models is an active research topic. In this paper, a mechanistic-based, data-driven computational framework is proposed for elastoplastic materials undergoing finite strain. Harnessing the physical insights from the existing model-based plasticity theory, multiplicative decomposition of deformation gradient and the coaxial relationship between the logarithmic trial elastic strain and the true stress is employed to perform stress-update, driven by two sets of the specifically measured one dimensional (1D) stress–strain data. The proposed approach, called MAP123-EPF, is used to solve several Boundary-Value Problems (BVPs) involving elastoplastic materials undergoing finite strains. Numerical results indicate that the proposed approach can predict the response of isotropic elastoplastic materials (characterized by the classical J2 plasticity model and the associative Drucker–Prager model) with good accuracy using numerically/experimentally generated data. The proposed approach circumvents the need for the several basic ingredients of a traditional finite strain computational plasticity model, such as an explicit yielding function, a hardening law and an appropriate objective stress rate. Demonstrative examples are shown and strengths and limitations of the proposed approach are discussed.

In this work, a unified AI-framework named Hierarchical Deep Learning Neural Network (HiDeNN) is proposed to solve challenging computational science and engineering problems with little or no available physics as well as with extreme computational demand. The detailed construction and mathematical elements of HiDeNN are introduced and discussed to show the flexibility of the framework for diverse problems from disparate fields. Three example problems are solved to demonstrate the accuracy, efficiency, and versatility of the framework. The first example is designed to show that HiDeNN is capable of achieving better accuracy than conventional finite element method by learning the optimal nodal positions and capturing the stress concentration with a coarse mesh. The second example applies HiDeNN for multiscale analysis with sub-neural networks at each material point of macroscale. The final example demonstrates how HiDeNN can discover governing dimensionless parameters from experimental data so that a reduced set of input can be used to increase the learning efficiency. We further present a discussion and demonstration of the solution for advanced engineering problems that require state-of-the-art AI approaches and how a general and flexible system, such as HiDeNN-AI framework, can be applied to solve these problems.

Thermal fluid coupled analysis is essential to enable an accurate temperature prediction in additive manufacturing. However, numerical simulations of this type are time-consuming, due to the high non-linearity, the underlying large mesh size and the small time step constraints. This paper presents a novel adaptive hyper reduction method for speeding up these simulations. The difficulties associated with non-linear terms for model reduction are tackled by designing an adaptive reduced integration domain. The proposed online basis adaptation strategy is based on a combination of a basis mapping, enrichment by local residuals and a gappy basis reconstruction technique. The efficiency of the proposed method will be demonstrated by representative 3D examples of additive manufacturing models, including single-track and multi-track cases.

Multiscale simulation of woven composites structure remains a challenge due to extremely expensive computational cost for solving the nonlinear woven Representative Volume Element (RVE). Recently, an effective and efficient Reduced Order modeling method, namely Self-consistent Clustering Analysis (SCA), is proposed to solve the RVE problem. In this work, the curse of computational cost in woven RVE problem is countered using the SCA, which maintains a considerable accuracy compared with the standard Finite Element Method (FEM). The Hill anisotropic yield surface is predicted efficiently using the woven SCA, which can accelerate the microstructure optimization and design of woven composites. Moreover, a two-scale FEM × SCA modeling framework is proposed for woven composites structure. Based on this framework, the complex behavior of the composite structures in macroscale can be predicted using microscale properties. Additionally, macroscale and mesoscale physical fields are captured simultaneously, which are hard, if not impossible, to observe using experimental methods. This will expedite the deformation mechanism investigation of composites. A numerical study is carried out for T-shaped hooking structures under cycle loading to illustrate these advantages.

In this paper, a mechanistic-based data-driven approach, MAP123-EP, is proposed for numerical analysis of elastoplastic materials. In this method, stress-update is driven by a set of one-dimensional stress–strain data generated by numerical or physical experiments under uniaxial loading. Numerical results indicate that combined with the classical strain-driven scheme, the proposed method can predict the mechanical response of isotropic elastoplastic materials (characterized by J2 plasticity model with isotropic/kinematic hardening and associated Drucker–Prager model) accurately without resorting to the typical ingredients of classical model-based plasticity, such as decomposing the total strain into elastic and plastic parts, as well as identifying explicit functional expressions of yielding surface and hardening curve. This mechanistic-based data-driven approach has the potential of opening up a new avenue for numerical analysis of problems where complex material behaviors cannot be described in explicit function/functional forms. The applicability and limitation of the proposed approach are also discussed.

Three different models with increased complexity to study the effects of hybridization on the tensile failure of hybrid composites are proposed. The first model is a model for dry bundles of fibres based on the statistics of fibre strength. The second is a model for composite materials based on the multiple fragmentation phenomenon. Lastly, a micromechanical numerical model is developed that considers a random distribution of fibres and takes into account the stochastic nature of fibre strength. This study aims to understand the controlling factors that lead to pseudo-ductility, as well as establish the sequence of failure mechanisms in hybrid composites under tensile loadings.

Additive manufacturing has risen to the top of research interest in advanced manufacturing in recent years due to process flexibility, achievability of geometric complexity, and the ability to locally modify and optimize materials. The present work is focused on providing an approach for incorporating thermodynamically consistent properties and microstructure evolution for non-equilibrium supercooling, as observed in additive manufacturing processes, into finite element analysis. There are two primary benefits of this work: (1) the resulting prediction is based on the material composition and (2) the nonlinear behavior caused by the thermodynamic properties of the material during the non-equilibrium solution is accounted for with extremely high resolution. The predicted temperature response and microstructure evolution for additively manufactured stainless steel 316L using standard handbook-obtained thermodynamic properties are compared with the thermodynamic properties calculated using the CALculation of PHAse Diagrams (CALPHAD) approach. Data transfer from the CALPHAD approach to finite element analysis is discussed.

Additive manufacturing (AM) methods for rapid prototyping of 3D materials (3D printing) have become increasingly popular with a particular recent emphasis on those methods used for metallic materials. These processes typically involve an accumulation of cyclic phase changes. The widespread interest in these methods is largely stimulated by their unique ability to create components of considerable complexity. However, modeling such processes is exceedingly difficult due to the highly localized and drastic material evolution that often occurs over the course of the manufacture time of each component. Final product characterization and validation are currently driven primarily by experimental means as a result of the lack of robust modeling procedures. In the present work, the authors discuss primary detrimental hurdles that have plagued effective modeling of AM methods for metallic materials while also providing logical speculation into preferable research directions for overcoming these hurdles. The primary focus of this work encompasses the specific areas of high-performance computing, multiscale modeling, materials characterization, process modeling, experimentation, and validation for final product performance of additively manufactured metallic components.

The multiresolution continuum theory is a higher order continuum theory where additional kinematic variables account for microstructural inhomogeneities at several distinct length scales. This can be particularly important for localization problems. The strength of this theory is that it can account for details in the microstructure of a material without using an extremely fine mesh. The present paper describes the implementation and verification of a 3D elastic–plastic multiresolution element based on an implicit time stepping algorithm. It is implemented in the general purpose finite element program FEAP. The mesh independency associated with the length scale parameter is examined and the convergence rate of the element is also evaluated.

An electron beam is a widely applied processing tool in welding and additive manufacturing applications. The heat source model of the electron beam acts as the basis of thermal simulations and predictions of the micro-structures and mechanical properties of the final products. While traditional volumetric and surface heat flux models were developed previously based on the observed shape of the molten pool produced by the beam, a new heat source model with a physically informed foundation has been established in this work. The new model was developed based on Monte Carlo simulations performed to obtain the distribution of absorbed energy through electron-atom collisions for an electron beam with a kinetic energy of 60 keV hitting a Ti---6Al---4V substrate. Thermal simulations of a moving electron beam heating a solid baseboard were conducted to compare the differences between the new heat source model, the traditional surface flux model and the volumetric flux model. Although the molten pool shapes with the three selected models were found to be similar, the predicted peak temperatures were noticeably different, which will influence the evaporation, recoil pressure and molten pool dynamics. The new heat source model was also used to investigate the influence of a static electron beam on a substrate. This investigation indicated that the new heat source model could scientifically explain phenomena that the surface and volumetric models cannot, such as eruption and explosion during electron beam processing.

A computational method is proposed for the analysis and prediction of instability (wrinkling or necking) of multilayered compressible plates and sheets made by metals or polymers under plane strain conditions. In previous works, a basic assumption (or a physical argument) that has been frequently made is that materials are incompressible to simplify mathematical derivations. To account for the compressibility of metals and polymers (the lower Poisson's ratio leads to the more compressible material), we propose a combined semi-numerical algorithm and finite element method for instability analysis. Our proposed algorithm is herein verified by comparing its predictions with published results in literature for thin films with polymer/metal substrates and for polymer/metal systems. The new combined method is then used to predict the effects of compressibility on instability behaviors. Results suggest potential utility for compressibility in the design of multilayered structures.

A continuing challenge in computational materials design is developing a model to link the microstructure of a material to its material properties in both an accurate and computationally efficient manner. In this paper, such a model is developed which uses image-based data from characterization studies combined with a newly developed self-consistent volume-integral micromechanics model (SVIM) for linear elastic material. It is observed that SVIM is able to capture the effective stress/strain distribution inside the inclusion, as well as effects of volume fraction and nearest inclusion distance on the effective properties of heterogeneous material. More importantly, SVIM can be applied to inclusions with arbitrary shape through discretizing the inclusion domain. For both 2-dimensional and 3-dimensional problems with circular and spherical inhomogeneities, SVIM's capability of predicting effective elastic properties is validated against experiments and direct numerical simulations using the finite element method. Finally, the effect of inclusion shape is predicted by SVIM.

Band structure characterizes the most important property of mechanical metamaterials. However, predicting the band structure of 3D metamaterials with complex microstructures through direct numerical simulation (DNS) is computationally inefficient due to the complexity of meshing. To overcome this issue, an extended finite element method (XFEM)-based method is developed to predict 3D metamaterial band structures. Since the microstructure and material interface are implicitly resolved by the level-set function embedded in the XFEM formulation, a non-conforming (such as uniform) mesh is used in the proposed method to avoid the difficulties in meshing complex geometries. The accuracy and mesh convergence of the proposed method have been validated and verified by studying the band structure of a spherical particle embedded in a cube and comparing the results with DNS. The band structures of 3D metamaterials with different microstructures have been studied using the proposed method with the same finite element mesh, indicating the flexibility of this method. This XFEM-based method opens new opportunities in design and optimization of mechanical metamaterials with target functions, e.g. location and width of the band gap, by eliminating the iterative procedure of re-building and re-meshing microstructures that is required by classical DNS type of methods.