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Abstract: Abstract The Nakajima test is a well-known material test from the steel and metal industry to determine the forming limit of sheet metal. It is demonstrated how FE2TI, our highly parallel scalable implementation of the computational homogenization method FE \(^2\) , can be used for the simulation of the Nakajima test. In this test, a sample sheet geometry is clamped between a blank holder and a die. Then, a hemispherical punch is driven into the specimen until material failure occurs. For the simulation of the Nakajima test, our software package FE2TI has been enhanced with a frictionless contact formulation on the macroscopic level using the penalty method. The appropriate choice of suitable boundary conditions as well as the influence of symmetry assumptions regarding the symmetric test setup are discussed. In order to be able to solve larger macroscopic problems more efficiently, the balancing domain decomposition by constraints (BDDC) approach has been implemented on the macroscopic level as an alternative to a sparse direct solver. To improve the computational efficiency of FE2TI even further, additionally, an adaptive load step approach has been implemented and different extrapolation strategies are compared. Both strategies yield a significant reduction of the overall computing time. Furthermore, a strategy to dynamically increase the penalty parameter is presented which allows to resolve the contact conditions more accurately without increasing the overall computing time too much. Numerically computed forming limit diagrams based on virtual Nakajima tests are presented. PubDate: 2021-11-01

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Abstract: Abstract A least squares recursive gradient meshfree collocation method is proposed for the superconvergent computation of structural vibration frequencies. The proposed approach employs the recursive gradients of meshfree shape functions together with smoothed shape functions in the context of least squares formulation, where both meshfree nodes and auxiliary points are taken as the collocation points. It turns out that this least squares formulation can effectively suppress the spurious modes arising from a direct meshfree collocation formulation using recursive gradients. Meanwhile, a detailed theoretical analysis with explicit frequency error measure is presented for the least squares recursive gradient meshfree collocation method in order to assess the frequency accuracy of structural vibrations. This analysis discloses the salient basis degree discrepancy issue regarding the frequency accuracy for the least squares meshfree collocation formulation, and it is shown that this issue can be essentially resolved by the proposed least squares recursive gradient meshfree collocation method. In fact, the proposed method leads to superconvergent vibration frequencies when odd degree basis functions are used, i.e., the frequency convergence rate is improved from \((p - 1)\) for the standard least squares meshfree collocation to \((p + 1)\) for the proposed approach in case of an odd pth degree basis function. This desirable frequency superconvergence of the proposed least squares recursive gradient meshfree collocation method is congruously demonstrated by numerical results. PubDate: 2021-11-01

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Abstract: Abstract A sequential nonlinear multiscale method for the simulation of elastic metamaterials subject to large deformations and instabilities is proposed. For the finite strain homogenization of cubic beam lattice unit cells, a stochastic perturbation approach is applied to induce buckling. Then, three variants of anisotropic effective constitutive models built upon artificial neural networks are trained on the homogenization data and investigated: one is hyperelastic and fulfills the material symmetry conditions by construction, while the other two are hyperelastic and elastic, respectively, and approximate the material symmetry through data augmentation based on strain energy densities and stresses. Finally, macroscopic nonlinear finite element simulations are conducted and compared to fully resolved simulations of a lattice structure. The good agreement between both approaches in tension and compression scenarios shows that the sequential multiscale approach based on anisotropic constitutive models can accurately reproduce the highly nonlinear behavior of buckling-driven 3D metamaterials at lesser computational effort. PubDate: 2021-11-01

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Abstract: Abstract Wrinkling instabilities occur when a stiff thin film bonded to an elastic substrate undergoes compression. Regardless of the nature of compression, this phenomenon has been extensively studied through local models based on classical continuum mechanics. However, the experimental behavior is not yet fully understood and the influence of nonlocal effects remains largely unexplored. The objective of this paper is to fill this gap from a computational perspective by investigating nonlocal wrinkling instabilities in a bilayered system. Peridynamics (PD), a nonlocal continuum formulation, serves as a tool to model nonlocal material behavior. This manuscript presents a methodology to precisely predict the critical conditions by employing an eigenvalue analysis. Our results approach the local solution when the nonlocality parameter, the horizon size, approaches zero. An experimentally observed influence of the boundaries on the wave pattern is reproduced with PD simulations which suggests nonlocal material behavior as a physical origin. The results suggest that the level of nonlocality of a material model has quantitative influence on the main wrinkling characteristics, while most trends qualitatively coincide with predictions from the local analytical solution. However, a relation between the film thickness and the critical compression is revealed that is not existent in the local theory. Moreover, an approach to determine the peridynamic material parameters across a material interface is established by introducing an interface weighting factor. This paper, for the first time, shows that adding a nonlocal perspective to the analysis of bilayer wrinkling by using PD can significantly advance our understanding of the phenomenon. PubDate: 2021-11-01

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Abstract: Abstract Neural networks (NN) have been studied and used widely in the field of computational mechanics, especially to approximate material behavior. One of their disadvantages is the large amount of data needed for the training process. In this paper, a new approach to enhance NN training with physical knowledge using constraint optimization techniques is presented. Specific constraints for hyperelastic materials are introduced, which include energy conservation, normalization and material symmetries. We show, that the introduced enhancements lead to better learning behavior with respect to well known issues like a small number of training samples or noisy data. The NN is used as a material law within a finite element analysis and its convergence behavior is discussed with regard to the newly introduced training enhancements. The feasibility of NNs trained with physical constraints is shown for data based on real world experiments. We show, that the enhanced training outperforms state-of-the-art techniques with respect to stability and convergence behavior within FE simulations. PubDate: 2021-11-01

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Abstract: Abstract Various biological processes such as transport of oxygen and nutrients, thrombus formation, vascular angiogenesis and remodeling are related to cellular/subcellular level biological processes, where mesoscopic simulations resolving detailed cell dynamics provide a key to understanding and identifying the cellular basis of disease. However, the intrinsic stochastic effects can play an important role in mesoscopic processes, while the time step allowed in a mesoscopic simulation is restricted by rapid cellular/subcellular dynamic processes. These challenges significantly limit the timescale that can be reached by mesoscopic simulations even with high-performance computing. To break this bottleneck and achieve a biologically meaningful timescale, we propose a multiscale parareal algorithm in which a continuum-based solver supervises a mesoscopic simulation in the time-domain. Using an iterative prediction-correction strategy, the parallel-in-time mesoscopic simulation supervised by its continuum-based counterpart can converge fast. The effectiveness of the proposed method is first verified in a time-dependent flow with a sinusoidal flowrate through a Y-shaped bifurcation channel. The results show that the supervised mesoscopic simulations of both Newtonian fluids and non-Newtonian bloods converge to reference solutions after a few iterations. Physical quantities of interest including velocity, wall shear stress and flowrate are computed to compare against those of reference solutions, showing a less than 1% relative error on flowrate in the Newtonian flow and a less than 3% relative error in the non-Newtonian blood flow. The proposed method is then applied to a large-scale mesoscopic simulation of microvessel blood flow in a zebrafish hindbrain for temporal acceleration. The three-dimensional geometry of the vasculature is constructed directly from the images of live zebrafish under a confocal microscope, resulting in a complex vascular network with 95 branches and 57 bifurcations. The time-dependent blood flow from heartbeats in this realistic vascular network of zebrafish hindbrain is simulated using dissipative particle dynamics as the mesoscopic model, which is supervised by a one-dimensional blood flow model (continuum-based model) in multiple temporal sub-domains. The computational analysis shows that the resulting microvessel blood flow converges to the reference solution after only two iterations. The proposed method is suitable for long-time mesoscopic simulations with complex fluids and geometries. It can be readily combined with classical spatial decomposition for further acceleration. PubDate: 2021-11-01

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Abstract: Abstract A method based on the Fast Fourier Transform is proposed to obtain the dispersion relation of acoustic waves in heterogeneous periodic media with arbitrary microstructures. The microstructure is explicitly considered using a voxelized Representative Volume Element (RVE). The dispersion diagram is obtained solving an eigenvalue problem for Bloch waves in Fourier space. To this aim, two linear operators representing stiffness and mass are defined through the use of differential operators in Fourier space. The smallest eigenvalues are obtained using the implicitly restarted Lanczos and the subspace iteration methods, and the required inverse of the stiffness operator is done using the conjugate gradient with a preconditioner. The method is used to study the propagation of acoustic waves in elastic polycrystals, showing the strong effect of crystal anistropy and polycrystaline texture on the propagation. It is shown that the method combines the simplicity of classical Fourier series analysis with the versatility of Finite Elements to account for complex geometries proving an efficient and general approach which allows the use of large RVEs in 3D. PubDate: 2021-11-01

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Abstract: Abstract This paper presents a risk-averse approach in the context of fail-safe topology optimization. The main novelty is the minimization of two risk functions quantifying the costs inherent to partial or full collapses, whose occurrence is considered as a source of uncertainty. This provides the designer with the flexibility to explicitly incorporate probabilistic information of occurrence of different structural failures, in contrast to the worst case approach, that penalizes all the damage configurations regardless their probability of occurrence. For the first time in the context of fail-safe topology optimization, a level-set method is employed. The level-set function is updated by means of a reaction–diffusion equation incorporating the topological derivative of the two risk-averse functions considered. Finally, the numerical experiments reveal the capability of the proposed formulations to yield redundant structures less sensitive to inherent losses of stiffness resulting from possible failures, whilst allowing designers to assume an acceptable level of risk. The benefits and drawbacks of the formulations proposed are compared against deterministic and fail-safe worst-case formulations. PubDate: 2021-11-01

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Abstract: Abstract The variational discrete element method developed in Marazzato et al. (Int J Numer Methods Eng, 121(23):5295–5319, 2020) for dynamic elasto-plastic computations is adapted to compute the deformation of elastic Cosserat materials. In addition to cellwise displacement degrees of freedom (dofs), cellwise rotational dofs are added. A reconstruction is devised to obtain \(P^1\) non-conforming polynomials in each cell and thus constant strains and stresses in each cell. The method requires only the usual macroscopic parameters of a Cosserat material and no microscopic parameter. Numerical examples show the robustness of the method for both static and dynamic computations in two and three dimensions. PubDate: 2021-11-01

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Abstract: Abstract We investigate the isogeometric analysis approach based on the extended Catmull–Clark subdivision for solving the PDEs on surfaces. As a compatible technique of NURBS, subdivision surfaces are capable of the refinability of B-spline techniques, and overcome the major difficulties of the interior parameterization encountered by the isogeometric analysis. The surface is accurately represented as the limit form of the extended Catmull–Clark subdivision, and remains unchanged throughout the h-refinement process. The solving of the PDEs on surfaces is processed on the space spanned by the Catmull–Clark subdivision basis functions. In this work, we establish the interpolation error estimates for the limit form of the extended Catmull–Clark subdivision function space on surfaces. We apply the results to develop the approximation estimates for solving multiple second-order PDEs on surfaces, which are the Laplace–Beltrami equation, the Laplace–Beltrami eigenvalue equation and the time-dependent Cahn–Allen equation. Numerical experiments confirm the theoretical results and are compared with the classical linear finite element method to demonstrate the performance of the proposed method. PubDate: 2021-11-01

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Abstract: Abstract This paper presents the computational stochastic homogenization of a heterogeneous 3D-linear anisotropic elastic microstructure that cannot be described in terms of constituents at microscale, as live tissues. The random apparent elasticity field at mesoscale is then modeled in a class of non-Gaussian positive-definite tensor-valued homogeneous random fields. We present an extension of previous works consisting of a novel probabilistic model to take into account uncertainties in the spectral measure of the random apparent elasticity field. A probabilistic analysis of the random effective elasticity tensor at macroscale is performed as a function of the level of spectrum uncertainties, which allows for studying the scale separation and the representative volume element size in a robust probabilistic framework. PubDate: 2021-11-01

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Abstract: Abstract Computational homogenization is a powerful tool allowing to obtain homogenized properties of materials on the macroscale from simulations of the underlying microstructure. The response of the microstructure is, however, strongly affected by variations in the microstructure geometry. In particular, we consider heterogeneous materials with randomly distributed non-overlapping inclusions, which radii are also random. In this work we extend the earlier proposed non-deterministic computational homogenization framework to plastic materials, thereby increasing the model versatility and overall realism. We apply novel soft periodic boundary conditions and estimate their effect in case of non-periodic material microstructures. We study macroscopic plasticity signatures like the macroscopic von-Mises stress and make useful conclusions for further constitutive modeling. Simulations demonstrate the effect of the novel boundary conditions, which significantly differ from the standard periodic boundary conditions, and the large influence of parameter variations and hence the importance of the stochastic modeling. PubDate: 2021-10-22

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Abstract: Abstract Moving boundaries and interfaces are commonly encountered in fluid flow simulations. For instance, fluid-structure interaction simulations require the formulation of the problem in moving domains, making the mesh distortion an issue of concern towards ensuring the accuracy of numerical model predictions. In this work, we propose a technique for the simultaneous mesh optimization and motion characterization. The mesh optimization/motion method introduced here is inspired by the mechanobiology of soft tissues, particularly those present in arterial walls, which feature an incredible capability to adapt to altered mechanical stimuli through adaptive mechanisms such as growth and remodeling. The proposed approach is in the framework of a low-distortion mesh moving method that is based on fiber-reinforced hyperelasticity and optimized zero-stress state. We adopt different reference configurations for the different constituents, namely ground substance and fibers. Hypothetical reference configurations are postulated for the different pieces of pseudo-material (the elements) as target shapes. Also, we modify the equilibrium equations using a volume-invariant strategy. Through the introduction of growth and remodeling adaptive processes we build an optimization algorithm which can attain an optimal configuration through a series of consecutive nonlinear optimizations steps. The remodeling mechanism allows to adapt the fiber deposition orientations, which become the driving force towards an homeostatic state, that is the optimal configuration. Also, a recruitment mechanism is introduced to selectively deal with initial highly distorted elements where high stresses develop due to the departure from the ideal configuration. We report 2D and 3D numerical experiments to show the application of this biologically-inspired mesh optimizer (BIMO) to simplicial finite element meshes. We also present additional numerical tests using BIMO as a mesh moving method. The results show that the proposed method performs satisfactorily, either as mesh optimizer and/or mesh motion strategy. PubDate: 2021-10-21

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Abstract: Abstract The Flory’s decomposition is an important mathematical tool used to write hyperelastic constitutive models. As far as the author’s knowledge goes, it has not been used to write plastic flow directions in elastoplastic models and this study is an opportunity to introduce this simple strategy in so important subject. Adopting this decomposition it is possible to write an alternative total Lagrangian elastoplastic framework for finite strains with simple implementation and good response. Using Flory’s decomposition, strains are split into one volumetric and two isochoric parts. The volumetric part is considered elastic along all strain range and isochoric parts are treated as elastoplastic, i.e., the isochoric plastic flow direction is directly defined by the Flory’s decomposition. Assuming this plastic flow direction it is not necessary to employ the classical Kröner-Lee multiplicative decomposition to consider elastic and plastic parts of finite strains. The proposed model is implemented in a 3D geometrical nonlinear positional FEM code and results are compared with literature experimental and numerical data for validation purposes and applications. PubDate: 2021-10-18

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Abstract: Abstract As an elementary mesh quality improvement technique, smoothing has been widely used in finite element (FE) analysis. Heuristic smoothing methods and optimization-based smoothing methods are the two main smoothing types. The former is efficient. However, it operates heuristically and may create low-quality elements. In contrast, optimization-based smoothing is very effective at improving mesh quality. However, it suffers from high computational cost since it calculates the optimal position of a free node iteratively. In this paper, we present a new smoothing method. The proposed method imitates the optimization-based smoothing based on a neural network, but it calculates the optimal position of a free node straightforwardly. Hence, the proposed method is more efficient than these optimization-based smoothing methods while being comparable in terms of mesh quality. We present various testing results to illustrate the effectiveness of the proposed method. PubDate: 2021-10-14

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Abstract: Abstract The computational methods for the shocks modeling would face two major challenges: (1) the severe damage with large deformations and (2) the intermittent waves. Peridynamics (PD) takes the integral form of its governing equation and shows exceeding advantages in modeling large deformation and severe damage. On the other hand, the propagation of intermittent wave within the PD based numerical system often experiences oscillatory instability. It can be attributed to the instability in time domain and the zero energy mode. Aiming for addressing such issues, the temporally stabilized PD methods are proposed in the present work. The stabilization force component is introduced and the general framework of stabilized PD methods is established. The formulation of the corresponding force state is proposed based on the features of the spurious oscillations. The case studies indicate that the stabilized PD methods are capable of effectively suppressing the nonphysical oscillations and are well-suited for the bond-based as well as the state-based PD formulations. PubDate: 2021-10-13

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Abstract: Abstract A three dimensional dendrite growth model is used to investigate the formation of shrinkage pores in nickel-based superalloy during the solidification process. A variety of simulations are performed with the temperature gradients and cooling rates under the casting and additive manufacturing conditions. The shrinkage pores are identified according to the localized liquid feeding condition. The investigation on casting not only validates the current model and methodology by comparing with a published X-ray imaging result but also uncovers more critical details, including the location preferences of shrinkage pores in the dendrite structure and the formation sequences of shrinkage pores in different stages, which have never been revealed in available literatures. After that, the effect of cooling rate on the pore formation is discussed for different mechanisms. In the investigation of additive manufacturing, the formation mechanisms of shrinkage pores are identified and well explain the shrinkage pores found in experimental results. Furthermore, the effects of the cooling rate and the new phases on the formation of shrinkage pores are discussed. PubDate: 2021-10-12

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Abstract: Abstract The impulsive differential equations are regarded as an optimal method to describe solute concentration fluctuation transport in unsteady flow field which are influenced by natural factors or human activities. The key difficulty of impulsive fractional-order system (IFS) in numerical discretization is that fractional-orders are different in different impulsive period. This paper proposes a double-scale-dependent mesh method considering the period memory, and makes a comparison with four collocation modes for the implict difference method. Furthermore, the stability and truncation error for graded meshes are estimated and analyzed. The analysis result reveals that the convergence rate mainly depends on the largest fractional order on the IFS. Numerical results show all graded meshes (producing the dense mesh at the early stage) provide better performance than uniform mesh. Meanwhile, the PDE cases show double-scale-dependent mesh is the most efficient numerical approximation method for the pulsation diffusion of contaminant in porous medium. PubDate: 2021-10-11

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Abstract: Abstract In this paper, a direct data-driven approach for the modeling of isotropic, tension–compression asymmetric, elasto-plastic materials is proposed. Our approach bypasses the conventional construction of explicit mathematical function-based elasto-plastic models, and the need for parameter-fitting. In it, stress update is driven directly by a set of stress–strain data that is generated from uniaxial tension and compression experiments (physical). Particularly, for compression experiments, digital image correlation and homogenization are combined to further improve modeling accuracy. Two representative tension–compression asymmetric materials, titanium alloy TC4ELI and high-density polyethylene, are chosen to illustrate the effectiveness and accuracy of our proposed approach. Results indicate that our data-driven approach can predict the mechanical response of elasto-plastic materials that exhibit tension–compression asymmetry, within the small deformation regime. This data-driven approach provides a practical way to model such materials directly from physical experimental data. Our current implementation is limited, however, by a small reduction to computational efficiency, when compared to typical function-based approaches. Moreover, our present formulation is focused on tension–compression asymmetric elasto-plastic materials that are isotropic. PubDate: 2021-10-10

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Abstract: Abstract A ubiquitous challenge in design space exploration or uncertainty quantification of complex engineering problems is the minimization of computational cost. A useful tool to ease the burden of solving such systems is model reduction. This work considers a stochastic model reduction method (SMR), in the context of polynomial chaos expansions, where low-fidelity (LF) samples are leveraged to form a stochastic reduced basis. The reduced basis enables the construction of a bi-fidelity (BF) estimate of a quantity of interest from a small number of high-fidelity (HF) samples. A successful BF estimate approximates the quantity of interest with accuracy comparable to the HF model and computational expense close to the LF model. We develop new error bounds for the SMR approach and present a procedure to practically utilize these bounds in order to assess the appropriateness of a given pair of LF and HF models for BF estimation. The effectiveness of the SMR approach, and the utility of the error bound are presented in three numerical examples. PubDate: 2021-10-09