Showing papers in "International Journal for Numerical Methods in Engineering in 2022"

A mechanics‐informed artificial neural network approach in data‐driven constitutive modeling

Abstract: A mechanics‐informed artificial neural network approach for learning constitutive laws governing complex, nonlinear, elastic materials from strain–stress data is proposed. The approach features a robust and accurate method for training a regression‐based model capable of capturing highly nonlinear strain–stress mappings, while preserving some fundamental principles of solid mechanics. In this sense, it is a structure‐preserving approach for constructing a data‐driven model featuring both the form‐agnostic advantage of purely phenomenological data‐driven regressions and the physical soundness of mechanistic models. The proposed methodology enforces desirable mathematical properties on the network architecture to guarantee the satisfaction of physical constraints such as objectivity, consistency (preservation of rigid body modes), dynamic stability, and material stability, which are important for successfully exploiting the resulting model in numerical simulations. Indeed, embedding such notions in a learning approach reduces a model's sensitivity to noise and promotes its robustness to inputs outside the training domain. The merits of the proposed learning approach are highlighted using several finite element analysis examples. Its potential for ensuring the computational tractability of multi‐scale applications is demonstrated with the acceleration of the nonlinear, dynamic, multi‐scale, fluid‐structure simulation of the supersonic inflation dynamics of a parachute system with a canopy made of a woven fabric.

<scp>Multiple‐GPU</scp> parallelization of three‐dimensional material point method based on single‐root complex

Abstract: As one of the arbitrary Lagrangian–Eulerian methods, the material point method (MPM) owns intrinsic advantages in simulation of large deformation problems by combining the merits of the Lagrangian and Eulerian approaches. Significant computational intensity is involved in the calculations of the MPM due to its very fine mesh needed to achieve a sufficiently high accuracy. A new multiple-GPU parallel strategy is developed based on a single-root complex architecture of the computer purely within a CUDA environment. Peer-to-Peer (P2P) communication between the GPUs is performed to exchange the information of the crossing particles and ghost element nodes, which is faster than the heavy send/receive operations between different computers through the infiniBand network. Domain decomposition is performed to split the whole computational task over the GPUs with a number of subdomains. The computations within each subdomain are allocated on a corresponding GPU using an enhanced “Particle-List” scheme to tackle the data race during the interpolation from associated particles to common nodes. The acceleration effect of the parallelization is evaluated with two benchmarks cases, mini-slump test after a dam break and cone penetration test in clay, where the maximum speedups with 1 and 8 GPUs are 88 and 604, respectively.

Optimum design and thermal modeling for 2D and 3D natural convection problems incorporating level set‐based topology optimization with body‐fitted mesh

Abstract: Passive heat sinks cooled by natural convection are reliable, compact, and low‐noise. They are widely used in telecommunication devices, LEDs, and so forth. This work builds upon the recent advancements in fluid topology optimization (TO) to present a case study of two‐ and three‐dimensional optimum design and thermal modeling for the natural convection problems using a reaction–diffusion equation (RDE)‐based level‐set method. To this end, first, a high‐fidelity thermal‐fluid model is constructed where the full Navier–Stokes equations are strongly coupled with the energy equation through the Boussinesq approximation. We benchmark our simulation solver against the experimental analysis and other numerical analysis methods. Next, we carefully investigate the flow behavior under different Grashof numbers using a fully transient simulation solver. Then, we revisit the RDE‐based level‐set TO methodology and construct an open‐source parallel TO framework. The main findings reveal that using the body‐fitted mesh adaptation, the proposed methodology can capture the explicit fluid–solid boundary and we are free of the continuation approach to penalize the design variable to the binary structure. A moderately large‐scale TO problem with 3.56×106 DOFs can be solved in parallel on a standard multi‐process platform. Our numerical implementation uses FreeFEM for finite element analysis, PETSc for distributed linear algebra, and Mmg for mesh adaptation. For comparison and for accessing our various techniques, a variety of 2D and 3D benchmarks are presented to support these remarkable features.

Molecular dynamics inferred transfer learning models for finite‐strain hyperelasticity of monoclinic crystals: Sobolev training and validations against physical constraints

Abstract: We present a machine learning framework to train and validate neural networks to predict the anisotropic elastic response of a monoclinic organic molecular crystal known as β$$ \beta $$ ‐HMX in the geometrical nonlinear regime. A filtered molecular dynamic (MD) simulations database is used to train neural networks with a Sobolev norm that uses the stress measure and a reference configuration to deduce the elastic stored free energy functional. To improve the accuracy of the elasticity tangent predictions originating from the learned stored free energy, a transfer learning technique is used to introduce additional tangential constraints from the data while necessary conditions (e.g., strong ellipticity, crystallographic symmetry) for the correctness of the model are either introduced as additional physical constraints or incorporated in the validation tests. Assessment of the neural networks is based on (1) the accuracy with which they reproduce the bottom‐line constitutive responses predicted by MD, (2) the robustness of the models measured by detailed examination of their stability and uniqueness, and (3) the admissibility of the predicted responses with respect to mechanics principles in the finite‐deformation regime. We compare the training efficiency of the neural networks under different Sobolev constraints and assess the accuracy and robustness of the models against MD benchmarks for β$$ \beta $$ ‐HMX.

Computational multiphase periporomechanics for unguided cracking in unsaturated porous media

Abstract: In this article, we formulate and implement a computational multiphase periporomechanics paradigm for unguided fracturing in unsaturated porous media assuming passive pore air pressure. The same governing equation for the solid phase applies on and off cracks. Crack formation in this framework is autonomous, requiring no prior estimates of crack topology. As a new contribution, an energy-based criterion for arbitrary crack formation is formulated using the peridynamic effective force state for unsaturated porous media. Unsaturated fluid flow in the fracture space is modeled in a simplified way in line with the nonlocal formulation of unsaturated fluid flow in the bulk. The formulated unsaturated fracturing periporomechanics is numerically implemented through an implicit fractional step algorithm in time and a two-phase mixed meshless method in space. The two-stage operator split converts the coupled periporomechanics problem into an undrained deformation and fracture problem and an unsaturated fluid flow in the deformed skeleton configuration. Numerical simulations of in-plane open and shear cracking are conducted to validate the accuracy and robustness of the fracturing unsaturated periporomechanics model. Then numerical examples of wing cracking and nonplanar cracking in unsaturated soil specimens are presented to demonstrate the efficacy of the proposed multiphase periporomechanics paradigm for unguided cracking in unsaturated porous media.

A nonperiodic Chebyshev spectral method avoiding penalization techniques for a class of nonlinear peridynamic models

Abstract: In the framework of elastodynamics, peridynamics is a nonlocal theory able to capture singularities without using partial derivatives. The governing equation is a second order in time partial integro‐differential equation. In this article, we focus on a one‐dimensional nonlinear model of peridynamics and propose a spectral method based on the Chebyshev polynomials to discretize in space. The main capability of the method is that it avoids the assumption of periodic boundary condition in the solution and can benefit of the use of the fast Fourier transform. We show its convergence and find that the method results to be very efficient in terms of accuracy and execution time with respect to spectral methods based on the Fourier trigonometric polynomials associated to a volume penalization technique.

A re‐evaluation of overshooting in time integration schemes: The neglected effect of physical damping in the starting procedure

Abstract: An important property of implicit time integration algorithms for structural dynamics is their tendency to “overshoot” the exact solution in the first few steps of the computed response due to high‐frequency components in the initial excitations. The typical analysis technique for overshooting involves the study of asymptotic response of the algorithm's first step in the limiting high frequency case. This article finds that the prior analysis of overshooting in much of the engineering literature is incomplete in that it neglects the effect of physical damping. With physical damping included, first‐order overshooting components enter into several well‐known time integration algorithms which were previously thought to exhibit zero‐order overshooting in displacement. The Newmark method, Wilson‐ θ method, Bazzi‐ ρ method, HHT‐ α method, WBZ‐ α method, and three parameter optimal/generalized‐ α method are analyzed, as well as the generalized single‐step single‐solve (GSSSS) framework which encompasses all of the prior schemes and other new and optimal algorithms and designs based upon the issues under consideration. The additional overshooting component is eliminated in the novel amended GSSSS V0 family (which is noteworthy and cannot be derived by conventional means), while the numerically dissipative schemes in the GSSSS U0 family (encompassing traditional methods such as the HHT‐ α method, WBZ‐ α method, and three parameter optimal/generalized‐ α method among other new algorithm designs) are shown to be irremediable as the additional overshooting component from physical damping enters into the second step of the response, which is a wholly new finding. Numerical verifications of the overshooting analysis are performed for SDOF and MDOF structures with and without physical damping, and practical recommendations are given for the solution of MDOF problems on the basis of algorithm design and selection for various initial conditions.

Virtual displacement based discontinuity layout optimization

Abstract: Discontinuity layout optimization (DLO) is a relatively new upper bound limit analysis method. Compared to classic topology optimization methods, aimed at obtaining the optimum design of a structure by considering its self‐weight, building cost, or bearing capacity, DLO optimizes the failure pattern of the structure under specific loading conditions and constraints by minimizing the dissipation energy. In this work, we present a modified DLO algorithm that contains all of the advantages of DLO. It is referred to virtual displacement‐based discontinuity layout optimization (VDLO). VDLO takes the stress state of a loaded structure as a snapshot and correspondingly provides the optimum failure pattern, which greatly extends the application potential of DLO. Numerical examples indicate the effectiveness and flexibility of VDLO. It is regarded as a highly promising supplemental tool for other numerical methods in element‐/node‐based frameworks.

The Schwarz alternating method for transient solid dynamics

Abstract: In our earlier work, we formulated the Schwarz alternating method as a means for concurrent multiscale coupling in finite deformation solid mechanics for quasi‐static problems. Herein, we advance this method for the study of transient dynamic multiscale solid mechanics problems where information is exchanged back and forth between small and large scales. The extension to dynamics relies on the notion of a global time stepper. Within each global time step, the subdomains are coupled by the standard Schwarz iterative process. Remarkably, each subdomain can use its own time step or even its own time integrator to advance its solution in time, provided that they synchronize at each global time step. We study the performance of the Schwarz method on several examples designed for this purpose. Our numerical experiments demonstrate that the method is capable of coupling regions with different mesh resolutions, different element types, and different time integration schemes (e.g., implicit and explicit), all without introducing any artifacts that afflict other coupling methods for transient dynamics. Finally, we apply the dynamic Schwarz alternating method to the simulation of a bolted joint subjected to dynamic loading, as a demonstration of the performance of the method in a realistic scenario.

Spectral Chebyshev method coupled with a high order continuation for nonlinear bending and buckling analysis of functionally graded sandwich beams

Abstract: The main objective of the present work is to couple the spectral Chebyshev differential quadrature method (SCDQM) to the high order continuation method (HOCM) which was proposed in previous works with several discretization techniques. This new approach (SCDQM‐HOCM) is proposed to analyze the nonlinear bending and buckling analysis of functionally graded sandwich beams. The originality of this work consists also to use a beam model which taken into account the nonlinear term neglected in several works of the literature. This term makes it possible to have the buckling and bending problems by using the classical Timoshenko model and to handle the boundary conditions that several works could not to take into account. A strong form of nonlinear equations is established based on the first order shear deformation theory of beams with the von‐Kármán kinematic hypothesis. Regarding functionally graded materials (FGM), two typical types are investigated: sandwich beam with FGM faces and ceramic core (Type‐A), and sandwich beam with FGM core and uniform faces (Type‐B). The accuracy and efficiency of the SCDQM‐HOCM compared with finite element method coupled with high order continuation method (FEM‐HOCM) are illustrated on numerical examples of the FGM beams and then the FG sandwich beams. Furthermore, a parametric study is led to carry out the influence of different skin‐core‐skin thickness ratios, span‐to‐height ratio, and volume fraction on the bending and buckling behavior of FG sandwich beams subjected to different loadings and various boundary conditions.

A simultaneous space‐time discretization approach to the inverse dynamics of geometrically exact strings

Abstract: A continuous space‐time Galerkin method is newly proposed for the numerical solution of inverse dynamics problems. The proposed space‐time finite element method is combined with servo‐constraints to partially prescribe the motion of the underlying mechanical system. The new approach to the feedforward control of infinite‐dimensional mechanical systems is motivated by the classical method of characteristics. In particular, it is shown that the simultaneous space‐time discretization is much better suited to solve the inverse dynamics problem than the semi‐discretization approach commonly applied in structural dynamics. Representative numerical examples dealing with elastic strings undergoing large deformations demonstrate the capabilities of the newly devised space‐time finite element method.

Flaw classification and detection in thin‐plate structures based on scaled boundary finite element method and deep learning

Abstract: The identification of internal structural flaws is an important research topic in structural health monitoring. At present, structural safety inspections based on nondestructive testing procedures mainly focus on qualitative analysis; hence, it is difficult to identify the scale of flaws quantitatively. In this article, an inversion model that can realize quantitative detection is proposed by combing the scaled boundary finite element method (SBFEM) with deep learning. First, the lamb wave propagation processes in thin structures containing flaws were simulated using the SBFEM, and the echo wave signal at an observation point was recorded and paired with the flaw information. Then, the paired SBFEM datasets were used to train the deep learning model employing a convolutional neural network. A flaw classification and identification model based on SBFEM datasets was established. For a thin structure with unknown flaw information, and by using the measured observation point signals, the established deep learning model can predict flaw information in real time. Finally, the model performance was verified using several numerical examples. The results show that the proposed model can accurately predict flaw information and is robust against noise; thus, it offers an advantage over existing nondestructive testing procedures and a development in the field of structural health monitoring.

A k‐means clustering machine learning‐based multiscale method for anelastic heterogeneous structures with internal variables

Abstract: A new machine‐learning based multiscale method, called k‐means FE 2 , is introduced to solve general nonlinear multiscale problems with internal variables and loading history‐dependent behaviors, without use of surrogate models. The macro scale problem is reduced by constructing clusters of Gauss points in a structure which are estimated to be in the same mechanical state. A k‐means clustering—machine learning technique is employed to select the Gauss points based on their strain state and sets of internal variables. Then, for all Gauss points in a cluster, only one micro nonlinear problem is solved, and its response is transferred to all integration points of the cluster in terms of mechanical properties. The solution converges with respect to the number of clusters, which is weakly depends on the number of macro mesh elements. Accelerations of FE 2 calculations up to a factor 50 are observed in typical applications. Arbitrary nonlinear behaviors including internal variables can be considered at the micro level. The method is applied to heterogeneous structures with local quasi‐brittle and elastoplastic behaviors and, in particular, to a nuclear waste package structure subject to internal expansions.

DeepPhysics: A physics aware deep learning framework for real‐time simulation

Abstract: Real-time simulation of elastic structures is essential in many applications, from computer-guided surgical interventions to interactive design in mechanical engineering. The finite element method is often used as the numerical method of reference for solving the partial differential equations associated with these problems. Deep learning methods have recently shown that they could represent an alternative strategy to solve physics-based problems. In this article, we propose a solution to simulate hyper-elastic materials using a data-driven approach, where a neural network is trained to learn the nonlinear relationship between boundary conditions and the resulting displacement field. We also introduce a method to guarantee the validity of the solution. In total, we present three contributions: an optimized data set generation algorithm based on modal analysis, a physics-informed loss function, and a hybrid Newton–Raphson algorithm. The method is applied to two benchmarks: a cantilever beam and a propeller. The results show that our network architecture trained with a limited amount of data can predict the displacement field in less than a millisecond. The predictions on various geometries, topologies, mesh resolutions, and boundary conditions are accurate to a few micrometers for nonlinear deformations of several centimeters of amplitude.

Computational coupled large‐deformation periporomechanics for dynamic failure and fracturing in variably saturated porous media

Abstract: The large‐deformation mechanics and multiphysics of continuous or fracturing partially saturated porous media under static and dynamic loads are significant in engineering and science. This article is devoted to a computational coupled large‐deformation periporomechanics paradigm assuming passive air pressure for modeling dynamic failure and fracturing in variably saturated porous media. The coupled governing equations for bulk and fracture material points are formulated in the current/deformed configuration through the updated Lagrangian–Eulerian framework. It is assumed that the horizon of a mixed material point remains spherical and its neighbor points are determined in the current configuration. As a significant contribution, the mixed interface/phreatic material points near the phreatic line are explicitly considered for modeling the transition from partial to full saturation (vice versa) through the mixed peridynamic state concept. We have formulated the coupled constitutive correspondence principle and stabilization scheme in the updated Lagrangian–Eulerian framework for bulk and interface points. We numerically implement the coupled large deformation periporomechanics through a fully implicit fractional‐step algorithm in time and a hybrid updated Lagrangian–Eulerian meshfree method in space. Numerical examples are presented to validate the implemented stabilized computational coupled large‐deformation periporomechanics and demonstrate its efficacy and robustness in modeling dynamic failure and fracturing in variably saturated porous media.

Phase‐field modeling of brittle fracture along the thickness direction of plates and shells

Abstract: The prediction of fracture in thin‐walled structures is decisive for a wide range of applications. Modeling methods such as the phase‐field method usually consider cracks to be constant over the thickness which, especially in load cases involving bending, is an imperfect approximation. In this contribution, fracture phenomena along the thickness direction of structural elements (plates or shells) are addressed with a phase‐field modeling approach. For this purpose, a new, so called “mixed‐dimensional” model is introduced, which combines structural elements representing the displacement field in the two‐dimensional shell midsurface with continuum elements describing a crack phase‐field in the three‐dimensional solid space. The proposed model uses two separate finite element discretizations, where the transfer of variables between the coupled two‐ and three‐dimensional fields is performed at the integration points which in turn need to have corresponding geometric locations. The governing equations of the proposed mixed‐dimensional model are deduced in a consistent manner from a total energy functional with them also being compared to existing standard models. The resulting model has the advantage of a reduced computational effort due to the structural elements while still being able to accurately model arbitrary through‐thickness crack evolutions as well as partly along the thickness broken shells due to the continuum elements. Amongst others, the higher accuracy as well as the numerical efficiency of the proposed model are tested and validated by comparing simulation results of the new model to those obtained by standard models using numerous representative examples.

A neural network‐enhanced reproducing kernel particle method for modeling strain localization

Abstract: Modeling the localized intensive deformation in a damaged solid requires highly refined discretization for accurate prediction, which significantly increases the computational cost. Although adaptive model refinement can be employed for enhanced effectiveness, it is cumbersome for the traditional mesh‐based methods to perform while modeling the evolving localizations. In this work, neural network‐enhanced reproducing kernel particle method (NN‐RKPM) is proposed, where the location, orientation, and shape of the solution transition near a localization is automatically captured by the NN approximation via a block‐level neural network (NN) optimization. The weights and biases in the blocked parameterization network control the location and orientation of the localization. The designed basic four‐kernel NN block is capable of capturing a triple junction or a quadruple junction topological pattern, while more complicated localization topological patters are captured by the superposition of multiple four‐kernel NN blocks. The standard RK approximation is then utilized to approximate the smooth part of the solution, which permits a much coarser discretization than the high‐resolution discretization needed to capture sharp solution transitions with the conventional methods. A regularization of the NN approximation is additionally introduced for discretization‐independent material responses. The effectiveness of the proposed NN‐RKPM is verified by a series of numerical verifications.

The extended peridynamic model for elastoplastic and/or fracture problems

Abstract: The strain‐based implementation method for the extended peridynamic model (XPDM) resolves the limitation of standard models where only a fixed Poisson's ratio can be achieved. In this contribution, the XPDM formulation is extended to include bond breakage and/or plasticity mechanisms. The elastoplastic and bond breakage algorithms are elaborated. To capture the fracture process, a shear mechanism is now incorporated to the bond breakage response, in addition to the standard stretching failure mode. It is shown that the shear mechanism is required to accurately reproduce mixed mode fracture behavior observed experimentally. To demonstrate the predictive ability of the strain‐based XPDM, a wide range of quasi‐static and dynamic loading conditions, for both brittle and elasto‐plastic materials, is considered against experimental results or practical engineering scenarios.

Deep learning‐based reduced order models for the real‐time simulation of the nonlinear dynamics of microstructures

Abstract: We propose a non-intrusive deep learning-based reduced order model (DL-ROM) capable of capturing the complex dynamics of mechanical systems showing inertia and geometric nonlinearities. In the first phase, a limited number of high fidelity snapshots are used to generate a POD-Galerkin ROM which is subsequently exploited to generate the data, covering the whole parameter range, used in the training phase of the DL-ROM. A convolutional autoencoder is employed to map the system response onto a low-dimensional representation and, in parallel, to model the reduced nonlinear trial manifold. The system dynamics on the manifold is described by means of a deep feedforward neural network that is trained together with the autoencoder. The strategy is benchmarked against high fidelity solutions on a clamped-clamped beam and on a real micromirror with softening response and multiplicity of solutions. By comparing the different computational costs, we discuss the impressive gain in performance and show that the DL-ROM truly represents a real-time tool which can be profitably and efficiently employed in complex system-level simulation procedures for design and optimization purposes.

Computational homogenization of higher‐order continua

Abstract: We introduce a novel computational framework for the multiscale simulation of higher-order continua that allows for the consideration of first-, second-, and third-order effects at both micro- and macro-level. In line with classical two-scale approaches, we describe the microstructure via representative volume elements that are attached at each integration point of the macroscopic problem. To take account of the extended continuity requirements of independent fields at micro- and macro-level, we discretize both scales via isogeometric analysis (IGA). As a result, we obtain an IGA 2 -method that is conceptually similar to the well-known FE 2 -method. We demonstrate the functionality and accuracy of this novel multiscale method by means of a series of multiscale simulations involving different kinds of higher-order continua.

Intelligent stiffness computation for plate and beam structures by neural network enhanced finite element analysis

Abstract: In the present study, new methods are proposed to replace the constitutive law and the entire tangent stiffness matrix in finite element analysis by artificial neural networks (ANNs). By combining the FEM with ANN, so‐called intelligent elements are developed. First, as an extension to recent trends in model‐based material law replacement, we introduce an additional loss term corresponding to the material stiffness. This training procedure is referred to as Sobolev training and ensures that the ANN learns both the function approximating the stress behavior and its first derivative (material stiffness). In a following step, we introduce three methods to replace the entire local stiffness matrix of an element by approximating its generalized force‐displacement relations. These methods also adopt ANNs with Sobolev training procedure to predict the mentioned quantities. Since neural networks (NN) are universal function approximators, they are used to extract the stiffness information for elements undergoing plastic deformation. The focus of this work is to establish a neural network‐based FEM framework (independent of NN topology) to introduce an enhanced‐material law and in a consequent step also approximate stiffness information of truss, beam, and plate elements taking physical non‐linear behavior into account.

Stress‐constrained multiscale topology optimization with connectable graded microstructures using the worst‐case analysis

Abstract: This article proposes a stress-constrained multiscale topology optimization approach with connectable graded microstructures. The proposed method includes two stages. In the first stage, the shape interpolation method is first employed to generate a series of connectable unit cells. Then the effective elasticity tensors are calculated by the numerical homogenization and XFEM. Besides, the worst-case analysis and stress correction factor are employed to predict the maximum microscopic stress of the unit cells under arbitrary loading conditions. Furthermore, reduced-order models for the stress correction factor and effective elasticity tensor are built to efficiently predict the mechanical properties of the unit cell with any specified volume fraction. In the second stage, stress-constrained topology optimization is employed to find the distribution of microstructures by using established reduced-order models. Except for applying approaches commonly used in the traditional stress-constrained topology optimization, the moving Heaviside function is also proposed to include the void material into optimization. Finally, a threshold projection scheme is performed to realize the design of multiscale structures. Two numerical examples are presented to validate the proposed method. In addition, because the worst-case analysis overestimates the structural stress, an evolutionary discrete optimization is employed to further explore the potential of the multiscale structures.

A stabilized cell‐based smoothed finite element method against severe mesh distortion in non‐Newtonian fluid–structure interaction

Abstract: Failure is an inevitability for the finite element method (FEM) to be performed on seriously degenerated four‐node quadrilateral (Q4) elements whose Jacobians become negative. This article proposes a stabilized selective integration formulation of the cell‐based smoothed FEM (CSFEM) for modeling non‐Newtonian fluid–structure interaction (NNFSI) where that acute mesh distortion arises. The Carreau–Yasuda and Oldroyd‐B fluids, respectively, interact with a geometrically nonlinear solid in NNFSI. The CSFEM is applied to discretize both the fluid and solid media in space. As the fluid mesh accounts for a substantial part of the discretized NNFSI system, seriously distorted Q4 elements are specified exclusively for the fluid field. In this case, each convex Q4 element is subdivided into four smoothing cells (SCs) whereas its concave counterpart is regarded as one single SC. In the meantime, the solid elements are treated as usual. To stabilize the one‐SC integral in smoothed Galerkin weak form, an hourglass control is subsequently introduced into the fluid momentum equations as well as the viscoelastic constitutive equation. The transient non‐Newtonian fluid equations are advanced with dual time steps of the characteristic‐based split scheme in time to further enhance the numerical stability and accuracy. After discussing other aspects of computational NNFSI, two benchmark problems are presented to demonstrate the effectiveness and performance of the developed methodology under the harsh environment.

Consistency of the full and reduced order models for evolve‐filter‐relax regularization of convection‐dominated, marginally‐resolved flows

Abstract: Numerical stabilization is often used to eliminate (alleviate) the spurious oscillations generally produced by full order models (FOMs) in under-resolved or marginally-resolved simulations of convection-dominated flows. In this article, we investigate the role of numerical stabilization in reduced order models (ROMs) of marginally-resolved, convection-dominated incompressible flows. Specifically, we investigate the FOM–ROM consistency, that is, whether the numerical stabilization is beneficial both at the FOM and the ROM level. As a numerical stabilization strategy, we focus on the evolve-filter-relax (EFR) regularization algorithm, which centers around spatial filtering. To investigate the FOM-ROM consistency, we consider two ROM strategies: (i) the EFR-noEFR, in which the EFR stabilization is used at the FOM level, but not at the ROM level; and (ii) the EFR-EFR, in which the EFR stabilization is used both at the FOM and at the ROM level. We compare the EFR-noEFR with the EFR-EFR in the numerical simulation of a 2D incompressible flow past a circular cylinder in the convection-dominated, marginally-resolved regime. We also perform model reduction with respect to both time and Reynolds number. Our numerical investigation shows that the EFR-EFR is more accurate than the EFR-noEFR, which suggests that FOM-ROM consistency is beneficial in convection-dominated, marginally-resolved flows.

Recursive grain remapping scheme for phase‐field models of additive manufacturing

Abstract: The phase‐field method is an attractive tool in modeling microstructural evolution due to rapid solidification under additive manufacturing conditions, but typical polycrystalline models are prone to grain coalescence. Grain tracking and remapping schemes can eliminate artificial coalescence and increase the number of grains and crystallographic orientations that may be considered in a simulation, but previous tracking and remapping schemes may not efficiently capture the complex grain morphologies that form during additive manufacturing. A new recursive scheme is derived from a binary tree of axis‐aligned bounding boxes that can efficiently represent columnar and irregularly shaped grains. We demonstrate the power of this approach by simulating microstructures with hundreds to thousands of grains and quantify the reduction in the number of order parameters required to represent the microstructure. The new scheme leads to orders of magnitude fewer computational resources as compared to the naïve paradigm of one grain per order parameter, and also offers a substantial improvement over algorithms derived from bounding spheres.

The reformulated Koiter–Newton method for thermal–mechanical buckling and postbuckling analysis of thin‐walled structures

Abstract: The reduced‐order modeling method, termed as the Koiter–Newton method, is reformulated to be applicable for geometrically nonlinear thermal–mechanical analysis of thin‐walled structures. The thermal load is treated as the independently unchanged load corresponding to the initial temperature field. The internal force space is expanded using the mechanical load, the thermal load, and the predefined perturbation loads. The thermal–mechanical reduced‐order model is constructed using the first to fourth‐order derivatives of strain energy with thermal effects in terms of the degrees of freedom (DOFs). An additional DOF related to the thermal load appears in the construction of reduced‐order model based on the novel Koiter theory. The path‐following scheme is proposed to make the method able to trace the entire geometrically nonlinear thermoelastic response. A much larger step size can be achieved benefiting from the favorable prediction of the reduced‐order model, compared to the classical Newton‐like methods. Both the temperature‐independent and temperature‐dependent material properties are considered. The thermal–mechanical buckling and postbuckling behaviors obtained by the proposed method are validated and discussed using various numerical examples.

A Robin‐Neumann scheme with quasi‐Newton acceleration for partitioned fluid‐structure interaction

Abstract: The Dirichlet–Neumann scheme is the most common partitioned algorithm for fluid‐structure interaction (FSI) and offers high flexibility concerning the solvers employed for the two subproblems. Nevertheless, it is not without shortcomings: to begin with, the inherent added‐mass effect often destabilizes the numerical solution severely. Moreover, the Dirichlet–Neumann scheme cannot be applied to FSI problems in which an incompressible fluid is fully enclosed by Dirichlet boundaries, as it is incapable of satisfying the volume constraint. In the last decade, interface quasi‐Newton methods have proven to control the added‐mass effect and substantially speed up convergence by adding a Newton‐like update step to the Dirichlet–Neumann coupling. They are, however, without effect on the incompressibility dilemma. As an alternative, the Robin‐Neumann scheme generalizes the fluid's boundary condition to a Robin condition by including the Cauchy stresses. While this modification in fact successfully tackles both drawbacks of the Dirichlet–Neumann approach, the price to be paid is a strong dependency on the Robin weighting parameter, with very limited a priori knowledge about good choices. This work proposes a strategy to merge these two ideas and benefit from their combined strengths. The resulting quasi‐Newton‐accelerated Robin‐Neumann scheme is compared to both Robin‐ and Dirichlet–Neumann variants. The numerical tests demonstrate that it does not only provide faster convergence, but also massively reduces the influence of the Robin parameter, mitigating the main drawback of the Robin‐Neumann algorithm.

Mapped shape optimization method for the rational design of cellular mechanical metamaterials under large deformation

Abstract: Soft cellular mechanical metamaterials (CMMs) have gained increasing attention due to their unique mechanical properties, especially when under large deformation. However, the strong nonlinearities and complex instabilities brought by the large deformation field is a critical challenge for the rational design of soft CMMs. In this work, we propose a mapped shape optimization method as a computational framework for inverse designs of soft CMMs. The core of this method is to introduce a fixed referential configuration. The geometric changes of the cellular structures are reflected by altering a differentiable shape map; and the deformation of the corresponding structures are determined by mapping the finite element computations to the referential configuration. Such formulation avoids the need to alter the background mesh and more importantly, provides an efficient way to compute the gradient of the objective functions with respect to the design variables via the adjoint method. The proposed method is of general purpose, and three distinct yet representative numerical examples are used to demonstrate the effectiveness of the method: optimizing unique overall mechanical properties, precise control of the onset of instability, and optimizing phononic band gaps. These examples cover a broad range of important engineering applications of soft CMMs.

Topology optimization of thermo‐elastic structures considering stiffness, strength, and temperature constraints over a wide range of temperatures

Abstract: This article proposes a thermo-elastic topology optimization with stiffness, strength, and temperature constraints involving a wide range of temperatures. The state equations for the linear elasticity and heat conduction are considered. Formulations involving minimum volume with compliance, stress and temperature limits under multiple thermal conditions are presented. The global stress and regional temperature metrics adopting the corrected aggregation function are used to evaluate the maximal stress and temperature, respectively. The stress stabilizing scheme is utilized to overcome the iteration oscillation stemming from highly nonlinear behavior of thermal stress constraints for multiple thermal conditions. To achieve clear optimized topologies under design-dependent loads, a continuation strategy for the relaxed Heaviside function is developed. Two 2D numerical examples are employed to illustrate the validity and practicability of the proposed approach. The results show that the optimized structures designed by a certain temperature may be damaged or have thermal deformations once the ambient temperature changes due to the thermal residual stress. The designs covering wide temperature range achieved by neglecting stiffness or strength constraints can result in stiffness reduction or strength failure. It is therefore imperative for the optimization to adopt a multi-physics model involving multi-constraints over a wide temperature range.