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We numerically investigate the hydrodynamic characteristics and analyze the instability mechanism of a two-dimensional inverted flag clamped by a cylinder. Two transition routes and a total of six kinds of solutions exist under this configuration for different diameters of cylinders due to complex bifurcations. Specifically, for small cylinders, the undeformed equilibrium transitions to static deformed equilibrium through a supercritical pitchfork bifurcation, which is judged by the weakly nonlinear analysis together with the global linear instability analysis. The instability mechanism is the lifting effect of the steady structure mode working at the leading edge of the elastic plate. For large cylinders, another unstable fluid mode (decoupled with structure mode) causes the disappearance of the static undeformed and deformed equilibrium, replaced by a small amplitude flapping. The structure mode and the flow mode mainly contribute to the growth of perturbations in plate and downstream cylinder regions respectively, which can excite multi-mode oscillating transition analyzed by proper orthogonal decomposition. Moreover, we find there is a critical diameter $D_c$ dividing the pitchfork bifurcation and Hopf bifurcation, and $D_c$ decreases with the increase of Reynolds number. Finally, we prove downstream vortex shedding can induce upward vortex-induced vibration of the plate and further improve the efficiency of energy transfer from the fluid to the structure during small-deflection flapping.

The advent of quantum computers, operating on entirely different physical principles and abstractions from those of classical digital computers, sets forth a new computing paradigm that can potentially result in game-changing efficiencies and computational performance. Specifically, the ability to simultaneously evolve the state of an entire quantum system leads to quantum parallelism and interference. Despite these prospects, opportunities to bring quantum computing to bear on problems of computational mechanics remain largely unexplored. In this work, we demonstrate how quantum computing can indeed be used to solve representative volume element (RVE) problems in computational homogenisation with polylogarithmic complexity of $\mathcal{O}((\log N)^c)$, compared to $\mathcal{O}(N^c)$ in classical computing. Thus, our quantum RVE solver attains exponential acceleration with respect to classical solvers, bringing concurrent multiscale computing closer to practicality. The proposed quantum RVE solver combines conventional algorithms such as a fixed-point iteration for a homogeneous reference material and the Fast Fourier Transform (FFT). However, the quantum computing reformulation of these algorithms requires a fundamental paradigm shift and a complete rethinking and overhaul of the classical implementation. We employ or develop several techniques, including the Quantum Fourier Transform (QFT), quantum encoding of polynomials, classical piecewise Chebyshev approximation of functions and an auxiliary algorithm for implementing the fixed-point iteration and show that, indeed, an efficient implementation of RVE solvers on quantum computers is possible. We additionally provide theoretical proofs and numerical evidence confirming the anticipated $\mathcal{O} \left ((\log N)^c \right)$ complexity of the proposed solver.

This paper concerns the study of optimal (supremum and infimum) uncertainty bounds for systems where the input (or prior) probability measure is only partially/imperfectly known (e.g., with only statistical moments and/or on a coarse topology) rather than fully specified. Such partial knowledge provides constraints on the input probability measures. The theory of Optimal Uncertainty Quantification allows us to convert the task into a constraint optimization problem where one seeks to compute the least upper/greatest lower bound of the system's output uncertainties by finding the extremal probability measure of the input. Such optimization requires repeated evaluation of the system's performance indicator (input to performance map) and is high-dimensional and non-convex by nature. Therefore, it is difficult to find the optimal uncertainty bounds in practice. In this paper, we examine the use of machine learning, especially deep neural networks, to address the challenge. We achieve this by introducing a neural network classifier to approximate the performance indicator combined with the stochastic gradient descent method to solve the optimization problem. We demonstrate the learning based framework on the uncertainty quantification of the impact of magnesium alloys, which are promising light-weight structural and protective materials. Finally, we show that the approach can be used to construct maps for the performance certificate and safety design in engineering practice.

This paper concerns the study of history dependent phenomena in heterogeneous materials in a two-scale setting where the material is specified at a fine microscopic scale of heterogeneities that is much smaller than the coarse macroscopic scale of application. We specifically study a polycrystalline medium where each grain is governed by crystal plasticity while the solid is subjected to macroscopic dynamic loads. The theory of homogenization allows us to solve the macroscale problem directly with a constitutive relation that is defined implicitly by the solution of the microscale problem. However, the homogenization leads to a highly complex history dependence at the macroscale, one that can be quite different from that at the microscale. In this paper, we examine the use of machine-learning, and especially deep neural networks, to harness data generated by repeatedly solving the finer scale model to: (i) gain insights into the history dependence and the macroscopic internal variables that govern the overall response; and (ii) to create a computationally efficient surrogate of its solution operator, that can directly be used at the coarser scale with no further modeling. We do so by introducing a recurrent neural operator (RNO), and show that: (i) the architecture and the learned internal variables can provide insight into the physics of the macroscopic problem; and (ii) that the RNO can provide multiscale, specifically FE2, accuracy at a cost comparable to a conventional empirical constitutive relation.

Deep learning surrogate models have shown promise in solving partial differential equations (PDEs). Among them, the Fourier neural operator (FNO) achieves good accuracy, and is significantly faster compared to numerical solvers, on a variety of PDEs, such as fluid flows. However, the FNO uses the Fast Fourier transform (FFT), which is limited to rectangular domains with uniform grids. In this work, we propose a new framework, viz., geo-FNO, to solve PDEs on arbitrary geometries. Geo-FNO learns to deform the input (physical) domain, which may be irregular, into a latent space with a uniform grid. The FNO model with the FFT is applied in the latent space. The resulting geo-FNO model has both the computation efficiency of FFT and the flexibility of handling arbitrary geometries. Our geo-FNO is also flexible in terms of its input formats, viz., point clouds, meshes, and design parameters are all valid inputs. We consider a variety of PDEs such as the Elasticity, Plasticity, Euler's, and Navier-Stokes equations, and both forward modeling and inverse design problems. Geo-FNO is $10^5$ times faster than the standard numerical solvers and twice more accurate compared to direct interpolation on existing ML-based PDE solvers such as the standard FNO.

Fully resolving dynamics of materials with rapidly-varying features involves expensive fine-scale computations which need to be conducted on macroscopic scales. The theory of homogenization provides an approach to derive effective macroscopic equations which eliminates the small scales by exploiting scale separation. An accurate homogenized model avoids the computationally-expensive task of numerically solving the underlying balance laws at a fine scale, thereby rendering a numerical solution of the balance laws more computationally tractable. In complex settings, homogenization only defines the constitutive model implicitly, and machine learning can be used to learn the constitutive model explicitly from localized fine-scale simulations. In the case of one-dimensional viscoelasticity, the linearity of the model allows for a complete analysis. We establish that the homogenized constitutive model may be approximated by a recurrent neural network (RNN) that captures the memory. The memory is encapsulated in the evolution of an appropriate finite set of internal variables, discovered through the learning process and dependent on the history of the strain. Simulations are presented which validate the theory. Guidance for the learning of more complex models, such as arise in plasticity, by similar techniques, is given.

In this paper, we propose physics-informed neural operators (PINO) that combine training data and physics constraints to learn the solution operator of a given family of parametric Partial Differential Equations (PDE). PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. Specifically, in PINO, we combine coarse-resolution training data with PDE constraints imposed at a higher resolution. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families and shows no degradation in accuracy even under zero-shot super-resolution, i.e., being able to predict beyond the resolution of training data. PINO uses the Fourier neural operator (FNO) framework that is guaranteed to be a universal approximator for any continuous operator and discretization-convergent in the limit of mesh refinement. By adding PDE constraints to FNO at a higher resolution, we obtain a high-fidelity reconstruction of the ground-truth operator. Moreover, PINO succeeds in settings where no training data is available and only PDE constraints are imposed, while previous approaches, such as the Physics-Informed Neural Network (PINN), fail due to optimization challenges, e.g., in multi-scale dynamic systems such as Kolmogorov flows.

The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We formulate the neural operator as a composition of linear integral operators and nonlinear activation functions. We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator. The proposed neural operators are also discretization-invariant, i.e., they share the same model parameters among different discretization of the underlying function spaces. Furthermore, we introduce four classes of efficient parameterization, viz., graph neural operators, multi-pole graph neural operators, low-rank neural operators, and Fourier neural operators. An important application for neural operators is learning surrogate maps for the solution operators of partial differential equations (PDEs). We consider standard PDEs such as the Burgers, Darcy subsurface flow, and the Navier-Stokes equations, and show that the proposed neural operators have superior performance compared to existing machine learning based methodologies, while being several orders of magnitude faster than conventional PDE solvers.

Chaotic systems are notoriously challenging to predict because of their sensitivity to perturbations and errors due to time stepping. Despite this unpredictable behavior, for many dissipative systems the statistics of the long term trajectories are governed by an invariant measure supported on a set, known as the global attractor; for many problems this set is finite dimensional, even if the state space is infinite dimensional. For Markovian systems, the statistical properties of long-term trajectories are uniquely determined by the solution operator that maps the evolution of the system over arbitrary positive time increments. In this work, we propose a machine learning framework to learn the underlying solution operator for dissipative chaotic systems, showing that the resulting learned operator accurately captures short-time trajectories and long-time statistical behavior. Using this framework, we are able to predict various statistics of the invariant measure for the turbulent Kolmogorov Flow dynamics with Reynolds numbers up to 5000.

The development of new materials and structures for extreme conditions including impact remains a continuing challenge despite steady advances. Design is currently accomplished using a sequential approach: an optimal material is first developed using the process-structure-properties paradigm, where performance is measured against a blended measure. Then, the structure is optimized while holding the material properties fixed. In this paper, we propose an alternative concurrent and goal-oriented optimization approach where both the material properties and the structure are optimized simultaneously against an overall system-wide performance measure. We develop a non-intrusive, high-performance computational framework based on DAKOTA and GMSH and use it to study the ballistic impact of a double-layer plate of strong AZ31B magnesium alloy and soft polyurea. We show that the proposed concurrent and goal-oriented optimization strategy can provide significant advantage over the traditional sequential optimization approach.

Magnesium (Mg) has the lowest density of all structural metals and has excellent potential for wide use in structural applications. While pure Mg has inferior mechanical properties; the addition of further elements at various concentrations has produced alloys with enhanced mechanical performance and corrosion resistance. An important consequence of adding such elements is that the saturated Mg matrix can locally decompose to form solute clusters and intermetallic particles, often referred to as precipitates. Controlling the shape, number density, volume fraction, and spatial distribution of solute clusters and precipitates significantly impacts the alloy's plastic response. Conversely, plastic deformation during thermomechanical processing can dramatically impact solute clustering and precipitation. In this paper, we first discuss how solute atoms, solute clusters, and precipitates can improve the mechanical properties of Mg alloys. We do so by primarily comparing three alloy systems: Mg-Al, Mg-Zn, and Mg-Y-based alloys. In the second part, we provide strategies for optimizing such microstructures by controlling nucleation and growth of solute clusters and precipitates during thermomechanical processing. In the third part, we briefly highlight how one can enable inverse design of Mg alloys by a more robust Integrated Computational Materials Design (ICMD) approach.

The recent decades have seen various attempts at accelerating the process of developing materials targeted towards specific applications. The performance required for a particular application leads to the choice of a particular material system whose properties are optimized by manipulating its underlying microstructure through processing. The specific configuration of the structure is then designed by characterizing the material in detail, and using this characterization along with physical principles in system level simulations and optimization. These have been advanced by multiscale modeling of materials, high-throughput experimentations, materials data-bases, topology optimization and other ideas. Still, developing materials for extreme applications involving large deformation, high strain rates and high temperatures remains a challenge. This article reviews a number of recent methods that advance the goal of designing materials targeted by specific applications.

The macroscopic properties of materials that we observe and exploit in engineering application result from complex interactions between physics at multiple length and time scales: electronic, atomistic, defects, domains etc. Multiscale modeling seeks to understand these interactions by exploiting the inherent hierarchy where the behavior at a coarser scale regulates and averages the behavior at a finer scale. This requires the repeated solution of computationally expensive finer-scale models, and often a priori knowledge of those aspects of the finer-scale behavior that affect the coarser scale (order parameters, state variables, descriptors, etc.). We address this challenge in a two-scale setting where we learn the fine-scale behavior from off-line calculations and then use the learnt behavior directly in coarse scale calculations. The approach draws from recent successes of deep neural networks, in combination with ideas from model reduction. The approach builds on the recent success of deep neural networks by combining their approximation power in high dimensions with ideas from model reduction. It results in a neural network approximation that has high fidelity, is computationally inexpensive, is independent of the need for a priori knowledge, and can be used directly in the coarse scale calculations. We demonstrate the approach on problems involving the impact of magnesium, a promising light-weight structural and protective material.

The macroscopic behavior of many materials is complex and the end result of mechanisms that operate across a broad range of disparate scales. An imperfect knowledge of material behavior across scales is a source of epistemic uncertainty of the overall material behavior. However, assessing this uncertainty is difficult due to the complex nature of material response and the prohibitive computational cost of integral calculations. In this paper, we exploit the multiscale and hierarchical nature of material response to develop an approach to quantify the overall uncertainty of material response without the need for integral calculations. Specifically, we bound the uncertainty at each scale and then combine the partial uncertainties in a way that provides a bound on the overall or integral uncertainty. The bound provides a conservative estimate on the uncertainty. Importantly, this approach does not require integral calculations that are prohibitively expensive. We demonstrate the framework on the problem of ballistic impact of a polycrystalline magnesium plate. Magnesium and its alloys are of current interest as promising light-weight structural and protective materials. Finally, we remark that the approach can also be used to study the sensitivity of the overall response to particular mechanisms at lower scales in a materials-by-design approach.

The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For partial differential equations (PDEs), neural operators directly learn the mapping from any functional parametric dependence to the solution. Thus, they learn an entire family of PDEs, in contrast to classical methods which solve one instance of the equation. In this work, we formulate a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture. We perform experiments on Burgers' equation, Darcy flow, and Navier-Stokes equation. The Fourier neural operator is the first ML-based method to successfully model turbulent flows with zero-shot super-resolution. It is up to three orders of magnitude faster compared to traditional PDE solvers. Additionally, it achieves superior accuracy compared to previous learning-based solvers under fixed resolution.

One of the main challenges in using deep learning-based methods for simulating physical systems and solving partial differential equations (PDEs) is formulating physics-based data in the desired structure for neural networks. Graph neural networks (GNNs) have gained popularity in this area since graphs offer a natural way of modeling particle interactions and provide a clear way of discretizing the continuum models. However, the graphs constructed for approximating such tasks usually ignore long-range interactions due to unfavorable scaling of the computational complexity with respect to the number of nodes. The errors due to these approximations scale with the discretization of the system, thereby not allowing for generalization under mesh-refinement. Inspired by the classical multipole methods, we propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Our multi-level formulation is equivalent to recursively adding inducing points to the kernel matrix, unifying GNNs with multi-resolution matrix factorization of the kernel. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.

The classical development of neural networks has been primarily for mappings between a finite-dimensional Euclidean space and a set of classes, or between two finite-dimensional Euclidean spaces. The purpose of this work is to generalize neural networks so that they can learn mappings between infinite-dimensional spaces (operators). The key innovation in our work is that a single set of network parameters, within a carefully designed network architecture, may be used to describe mappings between infinite-dimensional spaces and between different finite-dimensional approximations of those spaces. We formulate approximation of the infinite-dimensional mapping by composing nonlinear activation functions and a class of integral operators. The kernel integration is computed by message passing on graph networks. This approach has substantial practical consequences which we will illustrate in the context of mappings between input data to partial differential equations (PDEs) and their solutions. In this context, such learned networks can generalize among different approximation methods for the PDE (such as finite difference or finite element methods) and among approximations corresponding to different underlying levels of resolution and discretization. Experiments confirm that the proposed graph kernel network does have the desired properties and show competitive performance compared to the state of the art solvers.