Direct Finite Element Simulation Research Articles

An affine formulation of eigenstrain-based homogenization method and its application to polycrystal plasticity

Abstract Reduced order models (ROMs) are typically incorporated into concurrent multiscale approaches to allow for efficient nonlinear multiscale simulations and to alleviate high cost of direct nonlinear computational homogenization schemes. ROMs based on the ideas of transformation field analysis are among the most popular in the literature since they only require linear elastic simulations for model construction and typically have low number of degrees of freedom. However, these models have been shown to deliver overly stiff response in simulating wide range of materials. The present study focuses on mitigating this problem in the context of eigenstrain-based homogenization method (EHM) using instantaneous moduli information for polycrystal elastoviscoplasticity. For this purpose, a new EHM model is developed with the intention of using affine moduli for recomputation of the instantaneous localization tensors. The accuracy of the method is compared to the original EHM and direct crystal plasticity finite element simulations for several synthetic polycrystal microstructures, loading conditions and varying phase contrast. We show that the affine model delivers consistently softer response compared to the original EHM model. In particular, the affine model delivers notably more accurate response in the presence of high phase contrast. The affine EHM is able to capture local load redistribution through recomputation of the localization tensors.

A domain decomposing reduced-order method for solving the multi-domain transient heat conduction problems

In this article, a domain decomposition algorithm combined with a reduced-order model (ROM) is proposed to deal with the transient heat conduction problems in multi-domain structures. Using the domain decomposition algorithm and the finite element method (FEM), the original problem is decomposed into multiple non-overlapping subdomains, which can be solved independently under appropriate boundary conditions. However, additional inter-regional connected calculation will be brought through the simple geometry decomposition. Thus, the proper orthogonal decomposition (POD), a projection-based ROM technique, is utilized to further reduce the number of unknowns and the calculation time for each subdomain. To be more specific, the local POD bases of each subdomain are obtained by decomposing a snapshot matrix that gathered according to time by the application of the virtual boundary conditions on interfaces of the adjacent subdomains and other boundaries. Furthermore, the problems under different boundary conditions can be linked by the variable condensation technique on the interfaces, which enhances the efficiency of the numerical solver dramatically. Finally, the accuracy and efficiency of the proposed method are demonstrated by comparing the results calculated using both the proposed algorithm and the direct finite element simulation.

Sedimentation of a spheroidal particle in an elastoviscoplastic fluid

The sedimentation dynamics of a prolate spheroidal particle in an unbounded elastoviscoplastic (EVP) fluid is studied by direct finite element simulations under inertialess flow conditions. The Saramito–Giesekus constitutive equation is employed to model the suspending liquid. The arbitrary Lagrangian–Eulerian formulation is used to handle the particle motion. The sedimentation, lift, and angular velocities of spheroids with aspect ratio between 1 and 8 are computed as the initial orientation, Bingham, and Weissenberg numbers are varied. Similar to the purely viscoelastic case, a spheroid in an EVP fluid rotates up to align its major axis with the applied force. As the Bingham number increases, the settling rate monotonically reduces, while the angular velocity first increases and then decreases. The initial orientation has a relevant effect on the particle stoppage because of the different drag experienced by the spheroid as its orientation is varied. The yielded and unyielded regions around the spheroid reveal that, for particle oriented transversely to the force, the yielded envelope shrinks near the tips due to the fast spatial decay of the stresses, and unyielded regions appear along the surface of the particle, similar to the solid caps observed at the front and back of a sphere. Fluid plasticity enhances the negative wake phenomenon that is observed at Weissenberg numbers significantly lower than the purely viscoelastic case. The results of the drag correction coefficient for particles aligned with longest axis along the force are presented.

Numerical simulation of fiber orientation kinetics and rheology of fiber-filled polymers in uniaxial extension

During processing of fiber composites, the fiber-induced stresses influence the local flow fields, which, in turn, influence the stress distribution and the fiber orientation. Therefore, it is crucial to be able to predict the rheology of fiber-filled polymer composites. In this study, we investigate the fiber orientation kinetics and rheological properties of fiber composites in uniaxial extensional flow by comparing direct numerical finite element simulations to experimental results from our previous study [Egelmeers et al., “In-situ experimental investigation of fiber orientation kinetics during uniaxial extensional flow of polymer composites,” J. Rheol. 68, 171–185 (2023)]. In the simulations, fiber–fiber interactions only occur hydrodynamically and lubrication stresses are fully resolved by using adaptive meshing. We employed a 7-mode and a 5-mode viscoelastic Giesekus material model to describe the behavior of, respectively, a strain hardening low-density polyethylene (LDPE) matrix and a non-strain hardening linear LDPE matrix, and investigated the influence of the Weissenberg number, strain hardening, and fiber volume fraction on the fiber orientation kinetics. We found that none of these parameters influence the fiber orientation kinetics, which agrees with our experimental data. The transient uniaxial extensional viscosity of a fiber-filled polymer suspension is investigated by comparing finite element simulations to a constitutive model proposed by Hinch and Leal [“Time-dependent shear flows of a suspension of particles with weak Brownian rotations,” J. Fluid Mech. 57(4), 753–767 (1973)] and to experimental results obtained in our previous study [Egelmeers et al., “In-situ experimental investigation of fiber orientation kinetics during uniaxial extensional flow of polymer composites,” J. Rheol. 68, 171–185 (2023)]. The simulations describe the experimental data well. Moreover, high agreement is found for the transient viscosity as a function of fiber orientation between the model and the simulations. At high strains for high fiber volume fractions, however, the simulations show additional strain hardening, which we attribute to local changes in microstructure.

Deep Learning for Size-Agnostic Inverse Design of Random-Network 3D Printed Mechanical Metamaterials.

Practical applications of mechanical metamaterials often involve solving inverse problems aimed at finding microarchitectures that give rise to certain properties. The limited resolution of additive manufacturing techniques often requires solving such inverse problems for specific specimen sizes. Moreover, the candidate microarchitectures should be resistant to fatigue and fracture. Such a multi-objective inverse design problem is formidably difficult to solve but its solution is the key to real-world applications of mechanical metamaterials. Here, a modular approach titled "Deep-DRAM" that combines four decoupled models is proposed, including two deep learning (DL) models, a deep generative model based on conditional variational autoencoders, and direct finite element (FE) simulations. Deep-DRAM integrates these models into a framework capable of finding many solutions to the posed multi-objective inverse design problem based on random-network unit cells. Using an extensive set of simulations as well as experiments performed on 3D printed specimens, it is demonstrate that: 1) the predictions of the DL models are in agreement with FE simulations and experimental observations, 2) an enlarged envelope of achievable elastic properties (e.g., rare combinations of double auxeticity and high stiffness) is realized using the proposed approach, and 3) Deep-DRAM can provide many solutions to the considered multi-objective inverse design problem.

Sequential Deep Operator Networks (S-DeepONet) for predicting full-field solutions under time-dependent loads

Deep Operator Network (DeepONet), a recently introduced deep learning operator network, approximates linear and nonlinear solution operators by taking parametric functions (infinite-dimensional objects) as inputs and mapping them to solution functions in contrast to classical neural networks that need re-training for every new set of parametric inputs. In this work, we have extended the classical formulation of DeepONets by introducing sequential learning models like the gated recurrent unit (GRU) and long short-term memory (LSTM) in the branch network to allow for accurate predictions of the solution contour plots under parametric and time-dependent loading histories, known as Sequential DeepONets (S-DeepONets). Two example problems, one on transient heat transfer and the other on path-dependent plastic loading, were shown to demonstrate the capabilities of the new architectures compared to the benchmark DeepONet model with a feed-forward neural network (FNN) in the branch. Despite being more computationally expensive, the GRU- and LSTM-DeepONets lowered the prediction error by half (0.06% vs. 0.12%) compared to FNN-DeepONet in the heat transfer problem, and by 2.5 times (0.85% vs. 3%) in the plasticity problem. In all cases, the proposed DeepONets achieved a prediction R2 value of above 0.995, indicating superior accuracy. Results show that once trained, the proposed S-DeepONets can accurately predict the final full-field solution over the entire domain and are at least two orders of magnitude faster than direct finite element simulations, rendering it an accurate and robust surrogate model for rapid preliminary evaluations.

A Theoretical Contact Model for Rough Elastic Spheres

Abstract Investigating the contact of rough spheres is constructive to wide engineering applications and scientific research. However, the coexistence of global curvature and surface roughness brings difficulty in the analysis of this problem. In this work, we present a theoretical model for the contact between a rough elastic sphere and a rigid plane. As a fundament, the whole-range contact of nominally flat Gaussian rough surfaces is dealt with using a finite element method. With the load–area relation for the contact of nominally flat rough surfaces, the local real contact area of rough spherical contact is determined, in which the local pressure is given by the Hertzian model. Then, the total real contact area of rough spherical contact is obtained by integrating over the Hertzian contact area. It is found that the load–area relation for the contact of rough spheres is linear at light loads and converges to the result of smooth spherical contact as the load increases. This rough spherical contact model is validated through direct finite element simulations.

Study of Applicability of Triangular Impulse Response Function for Ultimate Strength of LNG Cargo Containment Systems under Sloshing Impact Loads

The LNG cargo containment system used in membrane-type LNG cargo tanks must have sufficient dynamic strength to withstand the impact of sloshing loads. However, performing direct dynamic nonlinear transient finite element assessments against design sloshing impact loads with different design specifications can be complicated and time-consuming. To address this, it is effective to use linear superposition methods, such as the triangular impulse response function (TIRF) method, to conduct dynamic transient FE assessments of LNG cargo containment systems. However, as LNG cargo containment systems have a high level of nonlinearities in terms of geometry, material, and boundary effects, it is necessary to evaluate the applicability of the TIRF method in advance. This study investigates the dynamic responses of an LNG cargo containment system using the TIRF method and compares the ultimate value of the structural responses and impulses with that obtained using direct dynamic nonlinear transient assessments. Based on a comparison of a series of FE analyses, the study proposes a design for the partial safety factors for calculating the ultimate bending and shear capacities of an LNG cargo containment system, taking into consideration the dynamic impact of sloshing loads using the TIRF method. Finally, the ultimate shear and bending capacities are calculated using the proposed method and compared with those obtained through direct dynamic nonlinear transient assessments. The results show that the proposed method provides conservative estimates against direct nonlinear finite element simulations, with a difference of around 10% for the mean minus two standard deviations. This approach can be practically applied for early basic design purposes in the shipbuilding industry.

An Automated Machine Learning Engine with Inverse Analysis for Seismic Design of Dams

This paper proposes a systematic approach for the seismic design of 2D concrete dams. As opposed to the traditional design method which does not optimize the dam cross-section, the proposed design engine offers the optimal one based on the predefined constraints. A large database of about 24,000 simulations is generated based on transient simulation of the dam-foundation-water system. The database includes over 150 various dam shapes, water levels, and material properties, as well as 160 different ground motion records. Automated machine learning (AutoML) is used to generate a surrogate model of dam response as a function of thirty variables. The accuracy of single- and multi-output surrogate models are compared, and the efficiency of the design engine for various settings is discussed. Next, a simple yet robust inverse analysis method is coupled with a multi-output surrogate model to design a hypothetical dam in the United States. Having the seismic hazard scenario, geological survey data, and also the concrete mix, the dam shape is estimated and compared to direct finite element simulation. The results show promising accuracy from the AutoML regression. Furthermore, the design shape from the inverse analysis is in good agreement with the design objectives and also the finite element simulations.

Strain‐Driven Thermal and Optical Instability in Silver/Amorphous‐Silicon Hyperbolic Metamaterials

AbstractHyperbolic metamaterials show exceptional optical properties, such as near‐perfect broadband absorption, due to their geometrically‐engineered optical anisotropy. Many of their proposed applications, such as thermophotovoltaics or radiative cooling, require high‐temperature stability. In this work, Ag/a‐Si multilayers are examined as a model system for the thermal stability of hyperbolic metamaterials. Using a combination of nanotomography, finite element simulations, and optical spectroscopy, the thermal and optical instability of the metamaterials is mapped. Although the thermal instability initiates at 300 °C, the hyperbolic dispersion persists up to 500 °C. Direct finite element simulations on tomographical data provide a route to decouple and evaluate interfacial and elastic strain energy contributions to the instability. Depending on stacking order the instability's driving force is either dominated by changes in anisotropic elastic strain energy due thermal expansion mismatch or by minimization of interfacial energy. These findings open new avenues to understand multilayer instability and pave the way to design hyperbolic metamaterials able to withstand high temperatures.

Non-linear homogenization of polymer composites with porous inclusions

We homogenize the mechanical properties of a two-phase polymer composite, made of a polyurethane matrix containing porous spherical silicone-based inclusions, and designed to be used for wave control as a local resonant insulator. First we consider a homogenized energy for the spherical inclusions, an expression obtained by an explicit method. Then, we couple inclusions and matrix by means of the second-order tangent method. Direct finite element simulations allow some comparisons with available experiments.

A new analytical model for the flattening of Gaussian rough surfaces

This paper advances an analytical incremental contact model for the purely elastic or elastic-perfectly plastic Gaussian rough surfaces. The contact is modelled by the accumulation of identical circular contacts with radius given by the total truncated area at varying heights divided by the contact patch number. The contact area-load relationship is derived analytically, showing approximate linearity for the contact fraction up to 10%. Good agreement is found between the new proposed model and the direct finite element simulations. To characterize the influence of plastic deformation, a dimensionless plasticity parameter is introduced as the ratio of yield strain to root mean square gradient of the rough surface. It is demonstrated that the general elastic-plastic contact response would approach to the limit of purely elastic as the plasticity parameter increases.

A new localized inverse identification method for high temperature testing under resistive heating: Application to the elastic‐viscoplastic behaviour of L‐PBF processed In718

AbstractThe mechanical behaviour of the nickel‐based superalloy In718, as processed from laser powder bed fusion (L‐PBF) additive manufacturing, is characterized at high temperature, from 800 to 1100°C. Samples built by L‐PBF are submitted to sequences combining uniaxial tensile load at different prescribed velocities, and relaxation steps of different durations, operated under resistive heating under vacuum, with a home‐developed testing machine. Tests are equipped with force evolution measurement, with infra‐red field imaging and thermocouples to capture the non‐uniform temperature distributions induced by resistive heating, and with digital image correlation to capture the non‐linear displacement fields. An inverse finite element strategy is developed to identify the parameters of a temperature‐dependent elastic‐viscoplastic behaviour model. The strategy is based on (i) direct finite element simulations of tests, (ii) a cost function expressing the distance between calculated and measured quantities, and (iii) a minimization algorithm. Direct numerical simulations are performed on a limited part of the working zone of samples, the zone of interest, with applied boundary conditions provided by DIC records and with an imposed temperature distribution provided by infra‐red imaging. The cost function is based on the force evolution only, for a series of different tests operated at different nominal temperatures. Optimum values of constitutive parameters are obtained by minimizing the cost function value, which is achieved with the home‐developed optimization platform MOOPI. Finally, the identified parameters are discussed with respect to the literature.

Elastic-Perfectly Plastic Contact of Rough Surfaces: An Incremental Equivalent Circular Model

Abstract In this paper, an incremental equivalent contact model is developed for elastic-perfectly plastic solids with rough surfaces. The contact of rough surface is modeled by the accumulation of circular contacts with varying radius, which is estimated from the geometrical contact area and the number of contact patches. For three typical rough surfaces with various mechanical properties, the present model gives accurate predictions of the load-area relation, which are verified by direct finite element simulations. An approximately linear load-area relation is observed for elastic-plastic contact up to a large contact fraction of 15%, and the influence of yield stress is addressed.

Turing instability analysis of a singular cross-diffusion problem

The population model by Busenberg and Travis is a paradigmatic model in ecology and tumor modeling because its ability to capture interesting phenomena such as segregation of populations. Its singular mathematical structure enforces the consideration of regularized problems to deduce properties as fundamental as the existence of solutions. In this article we perform a weakly nonlinear stability analysis of a general class of regularized problems to study the convergence of the instability modes in the limit of the regularization parameter. We demonstrate with some specific examples that the pattern formation observed in the regularized problems, with unbounded wave numbers, is not present in the limit problem because of the amplitude decay of the oscillations. We also check the results of the stability analysis with direct finite element simulations of the problem. For more information see https://ejde.math.txstate.edu/Volumes/2021/55/abstr.html

Nonlinear reduced order modeling for nonlinear response simulation of panels via Large Eddy Simulation

Supersonic vehicles are subject not only to aerodynamic heating, but also to different acoustics, one of which is aeroacoustic induced by pressure fluctuation of high speed flow. The state-of-art structure sonic fatigue analysis is conducted by using uniformly distributed random White Gaussian Noise (WGN). However, uniformly distributed excitation is usually not consistent with the actual situation, and the validity of the method needs further investigation. In the present study, a nonlinear reduced-order model (NLROM) is presented to compute nonlinear response of isotropic and composite plates. The NLROM is based on finite element (FE) model and is constructed by means of Galerkin projection of the full order system onto a small subspace. The input of the NLROM is aerodynamic and aeroacoustic loads, which are computed by Large Eddy Simulation (LES) and interpolated from aerodynamic grid to structure node by using Radial Basis Function (RBF). The nonlinear response of the isotropic and composite plates is computed by NLROM and compared with WGN. The results show that the NLROM offers nearly an order of magnitude speed up as compared with direct FE simulation and predicts shorter sonic fatigue life than WGN.

A micro-mechanical constitutive model for heterogeneous rocks with non-associated plastic matrix as implicit standard materials

In this work, we shall propose a new micro-mechanical constitutive model for the estimation of effective elastic-plastic behaviors of heterogeneous rocks. A bi-potential based incremental variational (BIV) approach is developed in order to take into account non-uniform local strain fields of constituents. The studied materials are composed of a non-associated and pressure sensitive plastic matrix, elastic inclusions and/or voids. For clarity, the local behavior of matrix is first described by an elastic perfectly-plastic model. Based on the bi-potential theory to dealing with non-associated plastic flow, the solid matrix is considered as pertaining to implicit standard materials (ISMs). The effective incremental bi-potential and macroscopic stress tensor are then estimated through an extension of the incremental variational method initially established for generalized standard materials(GSMs). The accuracy of the BIV model is verified by comparing the model’s predictions with the reference results obtained from direct finite element simulations. Furthermore, by assuming that the general formulation obtained for the perfectly plastic matrix remains valid for each loading increment, the BIV model is extended to considering that the solid matrix exhibits an isotropic hardening by using an explicit algorithm. The accuracy of the extended BIV model is also validated by a series of comparisons with the reference solutions obtained by direct finite element simulations for both inclusion-reinforced composites and porous materials. Both local and macroscopic responses are compared. As an example of application, the extended BIV model is finally applied to estimating the mechanical responses of typical claystone and sandstone under different loading paths.

Evaluation of complex Poisson’s ratio of aged asphalt field cores using direct tension test and finite element simulation

The aims of this study are to determine the complex Poisson’s ratio of asphalt mixtures based on linear viscoelastic theory and to evaluate the effect of the complex Poisson’s ratio on critical responses of asphalt pavements using finite element (FE) models. Direct tension tests were conducted on field core specimens, and horizontal and axial strains at the top and bottom of the field cores were recorded with loading time. The elastic–viscoelastic correspondence principle was utilized to compute the complex Poisson’s ratio including the magnitude and phase angle with loading frequency. It is shown that the magnitude and phase angle were time, temperature and aging dependent. To better illustrate the effect of the complex Poisson’s ratio on asphalt pavement responses, FE models with user-defined materials were developed to compute and compare pavement responses at critical locations with a constant Poisson’s ratio and a complex Poisson’s ratio. Besides, the effect of the aging time on the complex Poisson’s ratio was evaluated from pavement responses. The differences between these cases were highlighted, which indicated the importance of using the complex Poisson’s ratio in pavement design and performance evaluation.

Prediction of plastic yield surface for porous materials by a machine learning approach

The present paper focuses on the prediction of effective plastic yield surface of porous materials having a von Mises type solid matrix. Some typical explicit yield criteria obtained by different analytical homogenization methods are briefly reviewed and evaluated by using numerical results obtained from direct finite element simulations with different values of porosity. Each criterion has its own advantage and weakness. In order to get a better prediction, the Artificial Neuron Network (ANN) algorithm is adopted specially for the prediction of macroscopic yield stress of porous materials, seen as a regression problem with two input parameters and one output value. For the training purpose which is a key step in the ANN approach, new numerical results are presented in the present work with a wide range of porosity and of macroscopic stress triaxiality. Based on these data, the ANN approach is trained and it converges quickly. Then the ANN predictions are compared with numerical test data, a good agreement is found for all loading cases. Comparing with the existing yield criteria, the prediction given by the ANN approach is much more accuracy and easy to apply.

Instability driven surface patterns: Insights from direct three-dimensional finite element simulations

Surface patterns driven by mechanical instability commonly form in thin-film structures with a compliant substrate. In this study a practical three-dimensional (3D) finite element modeling approach is employed to directly simulate surface instabilities. It is based on our previously developed embedded imperfection technique in two-dimensions, and is now successfully implemented in 3D to capture sinusoidal wrinkling under uniaxial compression and the checkerboard pattern under equi-biaxial compression. The numerical approach overcomes typical challenges in computationally modeling thin-film buckling instability without the need of sophisticated treatment. The simulation results show that instability-controlled surface patterning is not an instantaneous process. A local disturbance in deformation originates from the imperfection site and propagates outward, eventually leading to a uniform pattern across the entire surface.