Computational multiscale solvers for continuum approaches

Computational multiscale analyses are currently ubiquitous in science and technology. Different problems of interest-e.g., mechanical, fluid, thermal, or electromagnetic-involving a domain with two or more clearly distinguished spatial or temporal scales, are candidates to be solved by using this technique. Moreover, the predictable capability and potential of multiscale analysis may result in an interesting tool for the development of new concept materials, with desired macroscopic or apparent properties through the design of their microstructure, which is now even more possible with the combination of nanotechnology and additive manufacturing. Indeed, the information in terms of field variables at a finer scale is available by solving its associated localization problem. In this work, a review on the algorithmic treatment of multiscale analyses of several problems with a technological interest is presented. The paper collects both classical and modern techniques of multiscale simulation such as those based on the proper generalized decomposition (PGD) approach. Moreover, an overview of available software for the implementation of such numerical schemes is also carried out. The availability and usefulness of this technique in the design of complex microstructural systems are highlighted along the text. In this review, the fine, and hence the coarse scale, are associated with continuum variables so atomistic approaches and coarse-graining transfer techniques are out of the scope of this paper

Computational Multiscale Solvers for Continuum Approaches

Computational multiscale analyses are currently ubiquitous in science and technology. Different problems of interest-e.g., mechanical, fluid, thermal, or electromagnetic-involving a domain with two or more clearly distinguished spatial or temporal scales, are candidates to be solved by using this technique. Moreover, the predictable capability and potential of multiscale analysis may result in an interesting tool for the development of new concept materials, with desired macroscopic or apparent properties through the design of their microstructure, which is now even more possible with the combination of nanotechnology and additive manufacturing. Indeed, the information in terms of field variables at a finer scale is available by solving its associated localization problem. In this work, a review on the algorithmic treatment of multiscale analyses of several problems with a technological interest is presented. The paper collects both classical and modern techniques of multiscale simulation such as those based on the proper generalized decomposition (PGD) approach. Moreover, an overview of available software for the implementation of such numerical schemes is also carried out. The availability and usefulness of this technique in the design of complex microstructural systems are highlighted along the text. In this review, the fine, and hence the coarse scale, are associated with continuum variables so atomistic approaches and coarse-graining transfer techniques are out of the scope of this paper.Abengoa Researc

A semi-implicit discrete-continuum coupling method for porous media based on the effective stress principle at finite strain

Abstract: A finite strain multiscale hydro-mechanical model is established via an extended Hill–Mandel condition for two-phase porous media. By assuming that the effective stress principle holds at unit cell scale, we established a micro-to-macro transition that links the micromechanical responses at grain scale to the macroscopic effective stress responses, while modeling the fluid phase only at the macroscopic continuum level. We propose a dual-scale semi-implicit scheme, which treats macroscopic responses implicitly and microscopic responses explicitly. The dual-scale model is shown to have good convergence rate, and is stable and robust. By inferring effective stress measure and poro-plasticity parameters, such as porosity, Biot’s coefficient and Biot’s modulus from micro-scale simulations, the multiscale model is able to predict effective poro-elasto-plastic responses without introducing additional phenomenological laws. The performance of the proposed framework is demonstrated via a collection of representative numerical examples. Fabric tensors of the representative elementary volumes are computed and analyzed via the anisotropic critical state theory when strain localization occurs. Keywords: Multiscale poromechanics; Semi-implicit scheme; Homogenization; Discrete-continuum coupling; DEM–FEM; Anisotropic critical stat

Process modeling of liquid composite molding processes

The polymer matrix composites (PMCs) are carving out a niche amid the keen market competition to replace the other material counterparts, e.g., metals. Due to the low weight and the corrosion resistance, the PMCs are wildly utilized from aerospace to automobile industries, both in the sectors of civilian and defense. To obtain high-quality products at low cost, the composites industry continues seeking for numerical simulation tools to predict the manufacturing processes instead of prototype testing and trials. Regarding the attractive liquid composite molding (LCM) process, it provides the possibility to produce net shape parts from composites. The challenges are how to identify the primary physics of LCM processes and develop mathematical models to represent them. Models need to be both accurate and efficient, which is not easy to achieve.To model LCM processes, we have one option that describes all physics at the macroscopic scale. The fundamental continuum mechanics principles, e.g., mass balance, momentum balance, energy balance, and entropy inequality, help us developing models. In this regard, the theory of porous media (TPM), which relies on the concept of volume fractions, can explain the problems of the saturated/unsaturated multi-phase materials. Darcy\u27s law describes the relation between the flow velocity and the pressure gradient, without accounting for the dual-scale flow. The air and resin compose the homogenized flow at the infusion stage. The existence of the capillary pressure influences the flow front, which has been revealed in this thesis. The finite element method is employed to solve for the homogenized flow pressure, and the degree of saturation with the staggered approach, especially the Streamline-Upwind/Petrov-Galerkin (SUPG) method is implemented to eradicate the stability problem.As to the fiber preform response, an assumption of shell kinematics is made to reduce the model from a full 3-D problem to a shell-like problem. Given this, an explicit formulation is obtained to express the normal directional stretch as a function of homogenized flow pressure. This model has been verified and validated by a resin infusion experiment. The model mimics the preform relaxation and lubrication mechanisms successfully and efficiently.So far, the works mentioned above aimed at the isothermal infusion stage. However, resin flow development, heat transfer, and resin curing are strongly interrelated during the whole LCM process. The holistic simulation of both the infusion stage and the curing stage is carried out in this thesis. Finally, we propose a system of coupled models to help process engineers to design and control process parameters by using virtual numerical experiments instead of the traditional trial-and-error approach

From multiscale modeling to metamodeling of geomechanics problems

In numerical simulations of geomechanics problems, a grand challenge consists of overcoming the difficulties in making accurate and robust predictions by revealing the true mechanisms in particle interactions, fluid flow inside pore spaces, and hydromechanical coupling effect between the solid and fluid constituents, from microscale to mesoscale, and to macroscale. While simulation tools incorporating subscale physics can provide detailed insights and accurate material properties to macroscale simulations via computational homogenizations, these numerical simulations are often too computational demanding to be directly used across multiple scales. Recent breakthroughs of Artificial Intelligence (AI) via machine learning have great potential to overcome these barriers, as evidenced by their great success in many applications such as image recognition, natural language processing, and strategy exploration in games. The AI can achieve super-human performance level in a large number of applications, and accomplish tasks that were thought to be not feasible due to the limitations of human and previous computer algorithms. Yet, machine learning approaches can also suffer from overfitting, lack of interpretability, and lack of reliability. Thus the application of machine learning into generation of accurate and reliable surrogate constitutive models for geomaterials with multiscale and multiphysics is not trivial. For this purpose, we propose to establish an integrated modeling process for automatic designing, training, validating, and falsifying of constitutive models, or "metamodeling". This dissertation focuses on our efforts in laying down step-by-step the necessary theoretical and technical foundations for the multiscale metamodeling framework. The first step is to develop multiscale hydromechanical homogenization frameworks for both bulk granular materials and granular interfaces, with their behaviors homogenized from subscale microstructural simulations. For efficient simulations of field-scale geomechanics problems across more than two scales, we develop a hybrid data-driven method designed to capture the multiscale hydro-mechanical coupling effect of porous media with pores of various different sizes. By using sub-scale simulations to generate database to train material models, an offline homogenization procedure is used to replace the up-scaling procedure to generate path-dependent cohesive laws for localized physical discontinuities at both grain and specimen scales. To enable AI in taking over the trial-and-error tasks in the constitutive modeling process, we introduce a novel “metamodeling” framework that employs both graph theory and deep reinforcement learning (DRL) to generate accurate, physics compatible and interpretable surrogate machine learning models. The process of writing constitutive models is simplified as a sequence of forming graph edges with the goal of maximizing the model score (a function of accuracy, robustness and forward prediction quality). By using neural networks to estimate policies and state values, the computer agent is able to efficiently self-improve the constitutive models generated through self-playing. To overcome the obstacle of limited information in geomechanics, we improve the efficiency in utilization of experimental data by a multi-agent cooperative metamodeling framework to provide guidance on database generation and constitutive modeling at the same time. The modeler agent in the framework focuses on evaluating all modeling options (from domain experts’ knowledge or machine learning) in a directed multigraph of elasto-plasticity theory, and finding the optimal path that links the source of the directed graph (e.g., strain history) to the target (e.g., stress). Meanwhile, the data agent focuses on collecting data from real or virtual experiments, interacts with the modeler agent sequentially and generates the database for model calibration to optimize the prediction accuracy. Finally, we design a non-cooperative meta-modeling framework that focuses on automatically developing strategies that simultaneously generate experimental data to calibrate model parameters and explore weakness of a known constitutive model until the strengths and weaknesses of the constitutive law on the application range can be identified through competition. These tasks are enabled by a zero-sum reward system of the metamodeling game and robust adversarial reinforcement learning techniques

A multiscale multi-permeability poroplasticity model linked by recursive homogenizations and deep learning

Many geological materials, such as shale, mudstone, carbonate rock, limestone and rock salt are multi-porosity porous media in which pores of different scales may co-exist in the host matrix. When fractures propagate in these multi-porosity materials, these pores may enlarge and coalesce and therefore change the magnitude and the principal directions of the effective permeability tensors. The pore-fluid inside the cracks and the pores of host matrix may interact and exchange fluid mass, but the difference in hydraulic properties of these pores often means that a single homogenized effective permeability tensor field is insufficient to characterize the evolving hydraulic properties of these materials at smaller time scale. Furthermore, the complexity of the hydro-mechanical coupling process and the induced mechanical and hydraulic anisotropy originated from the micro-fracture and plasticity at grain scale also makes it difficult to propose, implement and validate separated macroscopic constitutive laws for numerical simulations. This article presents a hybrid data-driven method designed to capture the multiscale hydro-mechanical coupling effect of porous media with pores of various different sizes. At each scale, data-driven models generated from supervised machine learning are hybridized with classical constitutive laws in a directed graph that represents the numerical models. By using sub-scale simulations to generate database to train material models, an offline homogenization procedure is used to replace the up-scaling procedure to generate cohesive laws for localized physical discontinuities at both grain and specimen scales. Through a proper homogenization procedure that preserves spatial length scales, the proposed method enables field-scale simulations to gather insights from meso-scale and grain-scale micro-structural attributes. This method is proven to be much more computationally efficient than the classical DEM–FEM or FEM2 approach while at the same time more robust and flexible than the classical surrogate modeling approach. Due to the usage of bridging-scale technique, the proposed model may provide multiple opportunities to incorporate different types of simulations and experimental data across different length scales for machine learning. Numerical issues will also be discussed

Computational Multiscale Methods

Many physical processes in material sciences or geophysics are characterized by inherently complex interactions across a large range of non-separable scales in space and time. The resolution of all features on all scales in a computer simulation easily exceeds today's computing resources by multiple orders of magnitude. The observation and prediction of physical phenomena from multiscale models, hence, requires insightful numerical multiscale techniques to adaptively select relevant scales and effectively represent unresolved scales. This workshop enhanced the development of such methods and the mathematics behind them so that the reliable and efficient numerical simulation of some challenging multiscale problems eventually becomes feasible in high performance computing environments

Multi-scale biomechanical study of transport phenomena in the intervertebral disc

Intervertebral disc (IVD) degeneration is primarily involved in back pain, a morbidity that strongly affects the quality of life of individuals nowadays. Lumbar IVDs undergo stressful mechanical loads while being the largest avascular tissues in our body: Mechanical principles alone cannot unravel the intricate phenomena that occur at the cellular scale which are fundamental for the IVD regeneration. The present work aimed at coupling biomechanical and relevant molecular transport processes for disc cells to provide a mechanobiological finite element framework for a deeper understanding of degenerative processes and the planning of regenerative strategies. Given the importance of fluid flow within the IVD, the influence of poroelastic parameters such as permeabilities and solid-phase stiffness of the IVD subtissues was explored. A continuum porohyperelastic material model was then implemented. The angles of collagen fibers embedded in the annulus fibrosus (AF) were calibrated. The osmotic pressure of the central nucleus pulposus (NP) was also taken into account. In a parallel study of the human vertebral bone, microporomechanics was used together with experimental ultrasonic tests to characterize the stiffness of the solid matrix, and to provide estimates of poroelastic coefficients. Fluid dynamics analyses and microtomographic images were combined to understand the fluid exchanges at the bone-IVD interface. The porohyperelastic model of a lumbar IVD with poroelastic vertebral layers was coupled with a IVD transport model of three solutes - oxygen, lactate and glucose - interrelated to reproduce the glycolytic IVD metabolism. With such coupling it was possible to study the effect of deformations, fluid contents, solid-phase stiffness, permeabilities, pH, cell densities of IVD subtissues and NP osmotic pressure on the solute transport. Moreover, cell death governed by glucose deprivation and lactate accumulation was included to explore the mechanical effect on cell viability. Results showed that the stiffness of the AF had the most remarkable role on the poroelastic behavior of the IVD. The permeability of the thin cartilage endplate and the NP stiffness were also relevant. The porohyperelastic model was shown to reproduce the local AF mechanics, provided the fiber angles were calibrated regionally. Such back-calculation led to absolute values of fibers angles and to a global IVD poromechanical behavior in agreement with experiments in literature. The inclusion of osmotic pressure in the NP also led to stress values under confined compression comparable to those measured in healthy and degenerated NP specimens. For the solid bone matrix, axial and transverse stiffness coefficients found experimentally in the present work agreed with universal mass density-elasticity relationships, and combined with continuum microporomechanics provided poroelastic coefficients for undrained and drained cases. The effective permeability of the vertebral bony endplate calculated with fluid dynamics was highly correlated with the porosity measured in microtomographic images. The coupling of transport and porohyperelastic models revealed a mechanical effect acting under large volume changes and high compliance, favored by healthy rather than degenerated IVD properties. Such effect was attributed to strain-dependent diffusivities and diffusion distances and was shown to be beneficial for IVD cells due to the load-dependent increases of glucose levels. Cell density, NP osmotic pressure and porosity were the most important parameters affecting the coupled mechano-transport of metabolites. This novel study highlights the restoration of both cellular and mechanical factors and has a great potential impact for novel designs of treatments focused on tissue regeneration. It also provides methodological features that could be implemented in clinical image-based tools and improve the multiscale understanding of the human spine mechanobiology

Numerical Model Reduction and Error Control for Computational Homogenization of Transient Problems

Multiscale modeling is a class of methods useful for numerical simulation of mechanics, in particular, when the microstructure of a material is of importance. The main advantage is the ability to capture the overall response, and, at the same time, account for processes and structures on the underlying fine scales. The FE2 procedure, "finite element squared", is one standard multiscale approach in which the constitutive relation is replaced with a boundary value problem defined on an Representative Volume Element (RVE) which contains the microscale features. The procedure thus involves the solution of finite element problems on two scales: one macroscopic problem and multiple RVE problems, typically one for each quadrature point in the macroscale mesh. While the solution of the independent RVE problems can be trivially parallelized it can still be computationally impractical to solve the two-scale problem, in particular for fine macroscale meshes. It is, therefore, of interest to investigate methods for reducing the computational cost of solving the individual RVE problems, while still having control of the accuracy.In this thesis the concept of Numerical Model Reduction (NMR) is applied for reducing the RVE problems by constructing a reduced spatial basis using Spectral Decomposition (SD) and Proper Orthogonal Decomposition. Computational homogenization of two different transient model problems have been studied: heat flow and consolidation. In both cases the RVE problem reduces to a system of ordinary differential equations, with dimension much smaller than of the finite element system.With the reduced basis and decreased computational time comes also loss of accuracy. Thus, in order to assess results from a reduced computation, it is useful to quantify the error. This thesis focuses solely on estimation of the error stemming from the reduced basis by assuming the fully resolved finite element solution to be exact, thereby ignoring e.g. time- and space-discretization errors. For the linear model problems guaranteed, fully computable, bounds are derived for the error in (i) a constructed "energy" norm and (ii) a user-defined quantity of interest within the realm of goal-oriented error estimation. In the non-linear case approximate, fully computable, bounds are derived based on the linearized error equation.In all cases an associated (non-physical) symmetrized variational problem in space-time is introduced as a "driver" for the estimate. From this residual-based estimates with low computational cost are obtained. In particular, no extra modes than the ones used for the reduced basis approximation are required. The performance of the estimator is demonstrated with numerical examples, and, for both the heat flow problem and the poroelastic problem, the error is overestimated by an order of magnitude, which is deemed acceptable given that the estimate is fully explicit and the extra cost is negligible