International Journal for Numerical Methods in Engineering最新文献

We present an algorithm for constructing efficient surrogate frequency-domain models of (nonlinear) parametric dynamical systems in a non-intrusive way. To capture the dependence of the underlying system on frequency and parameters, our proposed approach combines rational approximation and smooth interpolation. In the approximation effort, locally adaptive sparse grids are applied to effectively explore the parameter domain even if the number of parameters is modest or high. Adaptivity is also employed to build rational approximations that efficiently capture the frequency dependence of the problem. These two features enable our method to build surrogate models that achieve a user-prescribed approximation accuracy, without wasting resources in “oversampling” the frequency and parameter domains. Thanks to its non-intrusiveness, our proposed method, as opposed to projection-based techniques for model order reduction, can be applied regardless of the complexity of the underlying physical model. Notably, our algorithm for adaptive sampling can be used even when prior knowledge of the problem structure is not available. To showcase the effectiveness of our approach, we apply it in the study of an aerodynamic bearing. Our method allows us to build surrogate models that adequately identify the bearing's behavior with respect to both design and operational parameters, while still achieving significant speedups.

In topology optimization of transient problems, memory requirements and computational costs often become prohibitively large due to the backward-in-time adjoint equations. Common approaches such as the Checkpointing (CP) and Local-in-Time (LT) algorithms reduce memory requirements by dividing the temporal domain into intervals and by computing sensitivities on one interval at a time. The CP algorithm reduces memory by recomputing state solutions instead of storing them. This leads to a significant increase in computational cost. The LT algorithm introduces approximations in the adjoint solution to reduce memory requirements and leads to a minimal increase in computational effort. However, we show that convergence can be hampered using the LT algorithm due to errors in approximate adjoints. To reduce memory and/or computational time, we present two novel algorithms. The hybrid Checkpointing/Local-in-Time (CP/LT) algorithm improves the convergence behavior of the LT algorithm at the cost of an increased computational time but remains more efficient than the CP algorithm. The Parallel-Local-in-Time (PLT) algorithm reduces the computational time through a temporal parallelization in which state and adjoint equations are solved simultaneously on multiple intervals. State and adjoint fields converge concurrently with the design. The effectiveness of each approach is illustrated with two-dimensional density-based topology optimization problems involving transient thermal or flow physics. Compared to the other discussed algorithms, we found a significant decrease in computational time for the PLT algorithm. Moreover, we show that under certain conditions, due to the use of approximations in the LT and PLT algorithms, they exhibit a bias toward designs with short characteristic times. Finally, based on the required memory reduction, computational cost, and convergence behavior of optimization problems, guidelines are provided for selecting the appropriate algorithms.

Numerical modeling of immiscible two-phase flow in deformable porous media has become increasingly significant due to its applications in oil reservoir engineering, geotechnical engineering and many others. The coupling between two-phase flow and geomechanics gives rise to a major challenge to the development of physically consistent mathematical models and effective numerical methods. In this article, based on the concept of free energies and guided by the second law of thermodynamics, we derive a thermodynamically consistent mathematical model for immiscible two-phase flow in poro-viscoelastic media. The model uses the fluid and solid free energies to characterize the fluid capillarity and solid skeleton elasticity, so that it rigorously follows an energy dissipation law. The thermodynamically consistent formulation of the pore fluid pressure is naturally derived for the solid mechanical equilibrium equation. Additionally, the model ensures the mass conservation law for both fluids and solids. For numerical approximation of the model, we propose an energy stable and mass conservative numerical method. The method herein inherits the energy dissipation law through appropriate energy approaches and subtle treatments for the coupling between two phase saturations, the effective pore pressure and porosity. Using the locally conservative cell-centered finite difference methods on staggered grids with the upwind strategies for saturations and porosity, we construct the fully discrete scheme, which has the ability to conserve the masses of both fluids and solids as well as preserve the energy dissipation law at the fully discrete level. In particular, the proposed method is an unbiased algorithm, that is, treating the wetting phase, the non-wetting phase and the solid phase in the same way. Numerical results are also given to validate and verify the features of the proposed model and numerical method.

Sparse regression and feature extraction are the cornerstones of knowledge discovery from massive data. Their goal is to discover interpretable and predictive models that provide simple relationships among scientific variables. While the statistical tools for model discovery are well established in the context of linear regression, their generalization to nonlinear regression in material modeling is highly problem-specific and insufficiently understood. Here we explore the potential of neural networks for automatic model discovery and induce sparsity by a hybrid approach that combines two strategies: regularization and physical constraints. We integrate the concept of L_p regularization for subset selection with constitutive neural networks that leverage our domain knowledge in kinematics and thermodynamics. We train our networks with both, synthetic and real data, and perform several thousand discovery runs to infer common guidelines and trends: L₂ regularization or ridge regression is unsuitable for model discovery; L₁ regularization or lasso promotes sparsity, but induces strong bias that may aggressively change the results; only L₀ regularization allows us to transparently fine-tune the trade-off between interpretability and predictability, simplicity and accuracy, and bias and variance. With these insights, we demonstrate that L_p regularized constitutive neural networks can simultaneously discover both, interpretable models and physically meaningful parameters. We anticipate that our findings will generalize to alternative discovery techniques such as sparse and symbolic regression, and to other domains such as biology, chemistry, or medicine. Our ability to automatically discover material models from data could have tremendous applications in generative material design and open new opportunities to manipulate matter, alter properties of existing materials, and discover new materials with user-defined properties.

We recently proposed an efficient method facilitating the parametric study of a finite element mechanical simulation as a postprocessing step, that is, without the need to run multiple simulations: the a posteriori finite element method (APFEM). APFEM only requires the knowledge of the vertices of the parameter space and is able to predict accurately how the degrees of freedom of a simulation, i.e., nodal displacements, and other outputs of interests, for example, element stress tensors, evolve when simulation parameters vary within their predefined ranges. In our previous work, these parameters were restricted to material properties and loading conditions. Here, we extend the APFEM to additionally account for changes in the original geometry. This is achieved by defining an intermediary reference frame whose mapping is defined stochastically in the weak form. Subsequent deformation is then reached by correcting for this stochastic variation in the reference frame through multiplicative decomposition of the deformation gradient tensor. The resulting framework is shown here to provide accurate mechanical predictions for relevant applications of increasing complexity: (i) quantifying the stress concentration factor of a plate under uniaxial loading with one and two elliptical holes of varying eccentricities, and (ii) performing the stochastic homogenisation of a composite plate with uncertain mechanical properties and geometry inclusion. This extension of APFEM completes our original approach to account parametrically for geometrical alterations, in addition to boundary conditions and material properties. The advantages of this approach in our original work in terms of stochastic prediction, uncertainty quantification, structural and material optimisation and Bayesian inferences are all naturally conserved.

Modeling an aortic dissection represents a particular challenge from a numerical perspective, especially when it comes to the interaction between solid (aortic wall) and liquid (blood flow). The complexity of patient-specific simulations requires a variety of parameters, modeling assumptions and simplifications that currently hinder their routine use in clinical settings. We present a numerical framework that captures, among other things, the layer-specific anisotropic properties of the aortic wall, the non-Newtonian behavior of blood, patient-specific geometry, and patient-specific flow conditions. We compare hemodynamic indicators and stress measurements in simulations with increasingly complex material models for the vessel tissue ranging from rigid walls to anisotropic hyperelastic materials. We find that for the present geometry and boundary conditions, rigid wall simulations produce different results than fluid–structure interaction simulations. Considering anisotropic fiber contributions in the tissue model, stress measurements in the aortic wall differ, but shear stress-based biomarkers are less affected. In summary, the increasing complexity of the tissue model enables capturing more details. However, an extensive parameter set is also required. Since the simulation results depend on these modeling choices, variations can lead to different recommendations in clinical applications.

Machine learning (ML) and Deep learning (DL) are increasingly pivotal in the design of advanced metamaterials, seamlessly integrated with material or topology optimization. Their intrinsic capability to predict and interconnect material properties across vast design spaces, often computationally prohibitive for conventional methods, has led to groundbreaking possibilities. This paper introduces an innovative machine learning approach for the optimization of acoustic metamaterials, focusing on Multiresonant Layered Acoustic Metamaterial (MLAM), designed for targeted noise attenuation at low frequencies (below 1000 Hz). This method leverages ML to create a continuous model of the Representative Volume Element (RVE) effective properties essential for evaluating sound transmission loss (STL), and subsequently used to optimize the overall topology configuration for maximum sound attenuation using a Genetic Algorithm (GA). The significance of this methodology lies in its ability to deliver rapid results without compromising accuracy, significantly reducing the computational overhead of complete topology optimization by several orders of magnitude. To demonstrate the versatility and scalability of this approach, it is extended to a more intricate RVE model, characterized by a higher number of parameters, and is optimized using the same strategy. In addition, to underscore the potential of ML techniques in synergy with traditional topology optimization, a comparative analysis is conducted, comparing the outcomes of the proposed method with those obtained through direct numerical simulation (DNS) of the corresponding full 3D MLAM model. This comparative analysis highlights the transformative potential of this combination, particularly when addressing complex topological challenges with significant computational demands, ushering in a new era of metamaterial and component design.

The study of polymers has continued to gain substantial attention due to their expanding range of applications, spanning essential engineering fields to emerging domains like stretchable electronics, soft robotics, and implantable sensors. These materials exhibit remarkable properties, primarily stemming from their intricate polymer chain network, which, in turn, increases the complexity of precisely modeling their behavior. Especially for modeling elastomers and their fracture behavior, accurately accounting for the deformations of the polymer chains is vital for predicting the rupture in highly stretched chains. Despite the importance, many robust multiscale continuum frameworks for modeling elastomer fracture tend to simplify network deformations by assuming uniform behavior among chains in all directions. Recognizing this limitation, our study proposes a multiscale fracture model that accounts for the anisotropic nature of elastomer network responses. At the microscale, damage in the chains is assumed to be driven by both the chain's entropy and the internal energy due to molecular bond distortions. In order to bridge the stretching in the chains to the macroscale deformation, we employ the maximal advance path constraint network model, inherently accommodating anisotropic network responses. As a result, chains oriented differently can be predicted to exhibit varying stretch and, consequently, different damage levels. To drive macroscale fracture based on damages in these chains, we utilize the micromorphic regularization theory, which involves the introduction of dual local-global damage variables at the macroscale. The macroscale local damage variable is obtained through the homogenization of the chain damage values, resulting in the prediction of an isotropic material response. The macroscale global damage variable is subjected to nonlocal effects and boundary conditions in a thermodynamically consistent phase field continuum formulation. Moreover, the total dissipation in the system is considered to be mainly due to the breaking of the molecular bonds at the microscale. To validate our model, we employ the double-edge notched tensile test as a benchmark, comparing simulation predictions with existing experimental data. Additionally, to enhance our understanding of the fracturing process, we conduct uniaxial tensile experiments on a square film made up of polydimethylsiloxane (PDMS) rubber embedded with a hole and notches and then compare our simulation predictions with the experimental observations. Furthermore, we visualize the evolution of stretch and damage values in chains oriented along different directions to assess the predictive capacity of the model. The results are also compared with another existing model to evaluate the utility of our model in accurately simulating the fracture behavior of rubber-like materials.

The intractable multiscale constitutives and the high computational cost in direct numerical simulations are the bottlenecks in fracture analysis of heterogeneous materials. In an attempt to achieve a balance between accuracy and efficiency, we propose a mathematically rigorous phase-field model for multiscale fracture. Leveraging the phase-field theory, the difficulty of discrete-continuous coupling in conventional cross-scale crack propagation analysis is resolved by constructing a continuum description of the crack. Based on the asymptotic expansion, an equivalent two-field coupled boundary-value problem is well-defined, from which we rigorously derive the macroscopic equivalent parameters, including the equivalent elasticity tensor and the equivalent fracture toughness tensor. In our approach, both the displacement field and the phase-field are simultaneously expanded, allowing us to obtain a fracture toughness tensor with diagonal elements of the corresponding matrix controlling anisotropic fracture behavior and non-diagonal elements governing crack deflection. This enables multiscale finite element homogenization procedure to accurately reproduce microstructural information, and capture the crack deflection angle in anisotropic materials without any a priori knowledge. From the numerical results, the proposed multiscale phase-field method demonstrates a significant reduction in computation time with respect to full-field simulations. Moreover, the method accurately reproduces physical consistent anisotropic fracture of non-centrosymmetric porous media, and the experimentally consistent damage response of fiber-reinforced composites. This work fuses well-established mathematical homogenization theory with the cutting-edge fracture phase-field method, sparking a fresh perspective for the fracture of heterogeneous media.