Search

In this work we use algebraic dual spaces with a domain decomposition method to solve the Darcy equations. We define the broken Sobolev spaces and their finite dimensional counterparts. A global trace space is defined that connects the solution between the broken spaces. Use of algebraic dual spaces results in a sparse, metric-free representation of the incompressibility constraint, the pressure gradient term, and on the continuity constraint between the sub domains. To demonstrate this, we solve two test cases: (i) a manufactured solution case, and (ii) an industrial benchmark reservoir modelling problem SPE10. The results demonstrate that the dual spaces can be used for domain decomposition formulation, and despite having more unknowns, requires less simulation time compared to the continuous Galerkin formulation, without compromising on the accuracy of the solution.

Two-scale simulations are often employed to analyze the effect of the microstructure on a component's macroscopic properties. Understanding these structure–property relations is essential in the optimal design of materials for specific applications. However, these two-scale simulations are typically computationally expensive and infeasible in multi-query contexts such as optimization and material design. To make such analyses amenable, the microscopic simulations can be replaced by inexpensive-to-evaluate surrogate models. Such surrogate models must be able to handle microstructure parameters in order to be used for material design. A previous work focused on the construction of an accurate surrogate model for microstructures under varying loading and material parameters by combining proper orthogonal decomposition and Gaussian process regression. However, that method works only for a fixed geometry, greatly limiting the design space. This work hence focuses on extending the methodology to treat geometrical parameters. To this end, a method that transforms different geometries onto a parent domain is presented, that then permits existing methodologies to be applied. We propose to solve an auxiliary problem based on linear elasticity to obtain the geometrical transformations. The method has a good reducibility and can therefore be quickly solved for many different geometries. Using these transformations, combined with the nonlinear microscopic problem, we derive a fast-to-evaluate surrogate model with the following key features: (1) the predictions of the effective quantities are independent of the auxiliary problem, (2) the predicted stress fields automatically fulfill the microscopic balance laws and are periodic, (3) the method is non-intrusive, (4) the stress field for all geometries can be recovered, and (5) the sensitivities are available and can be readily used for optimization and material design. The proposed methodology is tested on several composite microstructures, where rotations and large variations in the shape of inclusions are considered. Finally, a two-scale example is shown, where the surrogate model achieves a high accuracy and significant speed up, thus demonstrating its potential in two-scale shape optimization and material design problems.

We present a general form of the gradient-enhanced damage theory and its numerical implementation using finite element method (FEM) in modeling quasi-static brittle crack growth in one- (1D), two- (2D) and three-dimensional (3D) bodies. Coupled equations of the equilibrium and a new implicit gradient damage formulation are introduced to govern the deformation of the solid and evolution of the damage. The resulting nonlocal damage evolution equation featuring the growth of diffusive crack is integrated with a characteristic length scale to eliminate the common mesh-bias issue in FEM implementation. In contrast to the traditional gradient-enhanced damage approaches, the nonlocal damage field here is defined as the primary variable of the damage evolution equation without interpolation mismatch between the displacement and nonlocal damage fields. For derivation of the material constitutive law and local damage parameter, a novel strain energy density (SED) function based on the energy limiter theory for brittle crack growth problems under small strain regime is introduced. To further improve the performance of the developed model, an initial SED threshold, which is used for determining the critical point when damage starts to initiate in the material, is integrated into the novel energy limiter theory. For preventing nonphysical failure in compression domains, the spectral decomposition technique for the strain tensor is adopted to split the reference SED. With integrating the energy limiter into the developed theory, unlike the conventional nonlocal damage theories where the interpretation of the length scale is still ambiguous, the developed nonlocal damage model defines the length scale parameter as the problem-dependent factor. The performance and ability of the proposed model are demonstrated via a set of representative numerical examples in 1D, 2D and 3D fracture problems.

This paper proposes a computational framework for the design optimization of stable structures under large deformations by incorporating nonlinear buckling constraints. A novel strategy for suppressing spurious buckling modes related to low-density elements is proposed. The strategy depends on constructing a pseudo-mass matrix that assigns small pseudo masses for DOFs surrounded by only low-density elements and degenerates to an identity matrix for the solid region. A novel optimization procedure is developed that can handle both simple and multiple eigenvalues wherein consistent sensitivities of simple eigenvalues and directional derivatives of multiple eigenvalues are derived and utilized in a gradient-based optimization algorithm - the method of moving asymptotes. An adaptive linear energy interpolation method is also incorporated in nonlinear analyses to handle the low-density elements distortion under large deformations. The numerical results demonstrate that, for systems with either low or high symmetries, the nonlinear stability constraints can ensure structural stability at the target load under large deformations. Post-analysis on the B-spline fitted designs shows that the safety margin, i.e., the gap between the target load and the 1st critical load, of the optimized structures can be well controlled by selecting different stability constraint values. Interesting structural behaviors such as mode switching and multiple bifurcations are also demonstrated., 77 pages, 44 Figures

Realistic microscale domains are an essential step towards making modern multiscale simulations more applicable to computational materials engineering. For this purpose, 3D computed tomography scans can be very expensive or technically impossible for certain materials, whereas 2D information can be easier obtained. Based on a single or three orthogonal 2D slices, the recently proposed differentiable microstructure characterization and reconstruction (DMCR) algorithm is able to reconstruct multiple plausible 3D realizations of the microstructure based on statistical descriptors, i.e., without the need for a training data set. Building upon DMCR, this work introduces a highly accurate two-stage reconstruction algorithm that refines the DMCR results under consideration of microstructure descriptors. Furthermore, the 2D-to-3D reconstruction is validated using a real computed tomography (CT) scan of a recently developed beta-Ti/TiFe alloy as well as anisotropic "bone-like" spinodoid structures. After a detailed discussion of systematic errors in the descriptor space, the reconstructed microstructures are compared to the reference in terms of the numerically obtained effective elastic and plastic properties. Together with the free accessibility of the presented algorithms in MCRpy, the excellent results in this study motivate interdisciplinary cooperation in applying numerical multiscale simulations for computational materials engineering.

Stochastic methods have recently been the subject of increased attention in Computational Mechanics for their ability to account for the stochasticity of both material parameters and geometrical features in their predictions. Among them, the Galerkin Stochastic Finite Element Method (GSFEM) was shown to be particularly efficient and able to provide accurate output statistics, although at the cost of intrusive coding and additional theoretical algebraic efforts. In this method, distributions of the stochastic parameters are used as inputs for the solver, which in turn outputs nodal displacement distributions in one simulation. Here, we propose an extension of the GSFEM—termed the A posteriori Finite Element Method or APFEM—where uniform distributions are taken by default to allow for parametric studies of the inputs of interest as a postprocessing step after the simulation. Doing so, APFEM only requires the knowledge of the vertices of the parameter space. In particular, one key advantage of APFEM is its use in the context of Bayesian inferences, where the random evaluations required by the Bayesian setting (usually done through Monte Carlo) can be done exactly without the need for further simulations. Finally, we demonstrate the potential of APFEM by solving forward models with parametric boundary conditions in the context of (i) metamaterial design and (ii) pitchfork bifurcation of the buckling of a slender structure; and demonstrate the flexibility of its use for Bayesian inference by (iii) inferring friction coefficient of a half plane in a contact mechanics problem and (iv) inferring the stiffness of a brain region in the context of cancer surgical planning.

We develop a fully data-driven model of anisotropic finite viscoelasticity using neural ordinary differential equations as building blocks. We replace the Helmholtz free energy function and the dissipation potential with data-driven functions that a priori satisfy physics-based constraints such as objectivity and the second law of thermodynamics. Our approach enables modeling viscoelastic behavior of materials under arbitrary loads in three-dimensions even with large deformations and large deviations from the thermodynamic equilibrium. The data-driven nature of the governing potentials endows the model with much needed flexibility in modeling the viscoelastic behavior of a wide class of materials. We train the model using stress-strain data from biological and synthetic materials including humain brain tissue, blood clots, natural rubber and human myocardium and show that the data-driven method outperforms traditional, closed-form models of viscoelasticity.

In this work, we introduce ShipHullGAN, a generic parametric modeller built using deep convolutional generative adversarial networks (GANs) for the versatile representation and generation of ship hulls. At a high level, the new model intends to address the current conservatism in the parametric ship design paradigm, where parametric modellers can only handle a particular ship type. We trained ShipHullGAN on a large dataset of 52,591 \textit{physically validated} designs from a wide range of existing ship types, including container ships, tankers, bulk carriers, tugboats, and crew supply vessels. We developed a new shape extraction and representation strategy to convert all training designs into a common geometric representation of the same resolution, as typically GANs can only accept vectors of fixed dimension as input. A space-filling layer is placed right after the generator component to ensure that the trained generator can cover all design classes. During training, designs are provided in the form of a shape-signature tensor (SST) which harnesses the compact geometric representation using geometric moments that further enable the inexpensive incorporation of physics-informed elements in ship design. We have shown through extensive comparative studies and optimisation cases that ShipHullGAN can generate designs with augmented features resulting in versatile design spaces that produce traditional and novel designs with geometrically valid and practically feasible shapes.

Highly accurate simulations of complex phenomena governed by partial differential equations (PDEs) typically require intrusive methods and entail expensive computational costs, which might become prohibitive when approximating steady-state solutions of PDEs for multiple combinations of control parameters and initial conditions. Therefore, constructing efficient reduced order models (ROMs) that enable accurate but fast predictions, while retaining the dynamical characteristics of the physical phenomenon as parameters vary, is of paramount importance. In this work, a data-driven, non-intrusive framework which combines ROM construction with reduced dynamics identification, is presented. Starting from a limited amount of full order solutions, the proposed approach leverages autoencoder neural networks with parametric sparse identification of nonlinear dynamics (SINDy) to construct a low-dimensional dynamical model. This model can be queried to efficiently compute full-time solutions at new parameter instances, as well as directly fed to continuation algorithms. These aim at tracking the evolution of periodic steady-state responses as functions of system parameters, avoiding the computation of the transient phase, and allowing to detect instabilities and bifurcations. Featuring an explicit and parametrized modeling of the reduced dynamics, the proposed data-driven framework presents remarkable capabilities to generalize with respect to both time and parameters. Applications to structural mechanics and fluid dynamics problems illustrate the effectiveness and accuracy of the proposed method.

We use a physics-informed neural network (PINN) to simultaneously model and optimize the flow around an airfoil to maximize its lift to drag ratio. The parameters of the airfoil shape are provided as inputs to the PINN and the multidimensional search space of shape parameters is populated with collocation points to ensure that the Navier–Stokes equations are approximately satisfied throughout. We use the fact that the PINN is automatically differentiable to calculate gradients of the lift-to-drag ratio with respect to the airfoil shape parameters. This allows us to optimize with the L-BFGS gradient-based algorithm, which is more efficient than non-gradient-based algorithms, without deriving an adjoint code. We train the PINN with adaptive sampling of collocation points, such that the accuracy of the solution improves as the solution approaches the optimal point. We demonstrate this on two examples: one that optimizes a single parameter, and another that optimizes eleven parameters. The method is successful and, by comparison with conventional CFD, we find that the velocity and pressure fields have small pointwise errors and that the method converges to optimal parameters. We find that different PINNs converge to slightly different parameters, reflecting the fact that there are many closely-spaced local minima when using stochastic gradient descent. This method can be applied relatively easily to other optimization problems and avoids the difficult process of writing adjoint codes. It is, however, more computationally expensive than adjoint-based optimization. As knowledge about training PINNs improves and hardware dedicated to neural networks becomes faster, this method of simultaneous training and optimization with PINNs could become easier and faster than using adjoint codes.