Monolithic Solution Research Articles

A phase field solution for modelling hyperelastic material and hydrogel fracture in ABAQUS

The phase field fracture model is attracting significant interest. To model fracture in hyperelastic material and hydrogel, we have implemented a robust two- and three-dimensional phase field method in the commercial finite element code ABAQUS/Standard. The method is based on the rate-independent variational principle of diffuse fracture and also exploits the analogy between the phase field evolution law and the heat transfer equation, enabling the use of Abaqus’ in-built features and sparing the need for defining user elements. The framework is shown to accommodate both staggered and monolithic solution schemes. The approach can properly simulate the fracture for both hyperelastic material and hydrogel under different boundary conditions. Several examples are provided to demonstrate the robustness of the method. The provided source codes and the tutorials make it easy for practicing engineers and scientists to model crack propagation in hyperelastic and gel materials.

Reduced Dislocation of GaAs Layer Grown on Ge-Buffered Si (001) Substrate Using Dislocation Filter Layers for an O-Band InAs/GaAs Quantum Dot Narrow-Ridge Laser.

The development of the low dislocation density of the Si-based GaAs buffer is considered the key technical route for realizing InAs/GaAs quantum dot lasers for photonic integrated circuits. To prepare the high-quality GaAs layer on the Si substrate, we employed an engineered Ge-buffer on Si, used thermal cycle annealing, and introduced filtering layers, e.g., strained-layer superlattices, to control/reduce the threading dislocation density in the active part of the laser. In this way, a low defect density of 2.9 × 107 cm−2 could be achieved in the GaAs layer with a surface roughness of 1.01 nm. Transmission electron microscopy has been applied to study the effect of cycling, annealing, and filtering layers for blocking or bending threading-dislocation into the InAs QDs active region of the laser. In addition, the dependence of optical properties of InAs QDs on the growth temperature was also investigated. The results show that a density of 3.4 × 1010 InAs quantum dots could be grown at 450 °C, and the photoluminescence exhibits emission wavelengths of 1274 nm with a fullwidth at half-maximum (FWHM) equal to 32 nm at room temperature. The laser structure demonstrates a peak at 1.27 μm with an FWHM equal to 2.6 nm under a continuous-wave operation with a threshold current density of ∼158 A/cm2 for a 4-μm narrow-ridge width InAs QD device. This work, therefore, paves the path for a monolithic solution for photonic integrated circuits when III−V light sources (which is required for Si photonics) are grown on a Ge-platform (engineered Ge-buffer on Si) for the integration of the CMOS part with other photonic devices on the same chip in near future.

Modeling and simulation of thin-walled piezoelectric energy harvesters immersed in flow using monolithic fluid–structure interaction

A monolithic numerical scheme for fluid–structure interaction with special interest in thin-walled piezoelectric energy harvesters driven by fluid is proposed. Employing a beam/shell model for the thin-walled structure in this particular application creates a FSI problem in which an ( n − 1 ) -dimensional structure is embedded in an n -dimensional fluid flow. This choice induces a strongly discontinuous pressure field along the moving fluid–solid interface. We overcome this challenge within a continuous finite element framework by a splitting-fluid-domain approach. The governing equations of the multiphysics problem are solved in a simultaneous fashion in order to reliably capture the main dynamic characteristics of the strongly-coupled system that involves a large deformation piezoelectric composite structure, an integrated electric circuit and an incompressible viscous fluid. The monolithic solution scheme is based on the weighted residuals method, with the boundary-fitted finite element method used for the discretization in space, and the generalized- α method for discretization in time. The proposed framework is evaluated against reference data of two popular FSI benchmark problems. Two additional numerical examples of flow-driven thin-walled piezoelectric energy harvesters demonstrate the feasibility of the framework to predict controlled cyclic response and limit-cycle oscillations and thus the power output in typical operational states of this class of energy harvesting devices. • A fully monolithic model tailored to flow-driven thin-walled piezoelectric energy harvesters (PEHs). • A novel method to tackle the strongly discontinuous pressure field of the fluid field. • The computational framework is validated against classical FSI benchmark problems. • The importance of optimized PEH electrode coverage is revealed by two new benchmarks. • Reference implementation using the open-source finite element framework FEniCS.

Phase field modeling of brittle fracture in large-deformation solid shells with the efficient quasi-Newton solution and global–local approach

To efficiently predict the crack propagation in thin-walled structures, a global–local approach for phase field modeling using large-deformation solid shell finite elements considering the enhanced assumed strain (EAS) and the assumed natural strain (ANS) methods for the alleviation of locking effects is developed in this work. Aiming at tackling the poor convergence performance of standard Newton schemes, a quasi-Newton (QN) scheme is proposed for the solution of coupled governing equations stemming from the enhanced assumed strain shell formulation in a monolithic manner. The excellent convergence performance of this QN monolithic scheme for the multi-field shell formulation is demonstrated through several paradigmatic boundary value problems, including single edge notched tension and shear, fracture of cylindrical structure under mixed loading and fatigue induced crack growth. Compared with the popular alternating minimization (AM) or staggered solution scheme, it is also found that the QN monolithic solution scheme for the phase field modeling using enhanced strain shell formulation is very efficient without the loss of robustness, and significant computational gains are observed in all the numerical examples. In addition, to further reduce the computational cost in fracture modeling of large-scale thin-walled structures, a specific global–local phase field approach for solid shell elements in the 3D setting is proposed, in which the full displacement-phase field problem is considered at the local level, while addressing only the elastic problem at the global level. Its capability is demonstrated by the modeling of a cylindrical structure subjected to both static and fatigue cyclic loading conditions, which can be appealing to industrial applications.

On the computational treatment of fully coupled crystal plasticity slip and martensitic transformation constitutive models at finite strains

AbstractIn this article, four numerical techniques are formulated to improve the reliability and numerical efficiency of stress update algorithms of crystal‐plasticity‐like phenomena at finite strains. The resulting algorithmic setting is especially relevant for structural analyses of polycrystals and the multiscale evaluation of anisotropic microstructures in metallic materials. These techniques are exemplified with a constitutive model, which couples crystallographic slip and martensitic transformation deformation mechanisms with a viscous regularization. The model is characterized by a set of highly coupled equations expressed in a single system and solved by a monolithic solution procedure. The first technique employs a sub‐stepping procedure to generate a better initial guess for the Newton–Raphson scheme used in the iterative solution of the equilibrium problem at the Gauss quadrature point. Then, a logarithmic discretization of the exponential parameter of the viscoplastic law is proposed to approximate the target value of the exponential parameter incrementally in the return mapping algorithm. A strategy to remove the strain‐rate dependence and reduce the necessary viscoplastic parameters is also presented to help with the stiff equations that arise in the rate‐independent limit. Finally, an efficient strategy is suggested to complete the transformation process when the full martensitic transformation is approached. These techniques can be easily used in any combination and dramatically improve the model's efficiency by enabling more significant incremental steps to be used within the monolithic solution procedure. A thorough assessment of their impact is shown in a series of ablation studies.

Hybrid of monolithic and staggered solution techniques for the computational analysis of fracture, assessed on fibrous network mechanics

The computational analysis of fiber network fracture is an emerging field with application to paper, rubber-like materials, hydrogels, soft biological tissue, and composites. Fiber networks are often described as probabilistic structures of interacting one-dimensional elements, such as truss-bars and beams. Failure may then be modeled as strong discontinuities in the displacement field that are directly embedded within the structural finite elements. As for other strain-softening materials, the tangent stiffness matrix can be non-positive definite, which diminishes the robustness of the solution of the coupled (monolithic) two-field problem. Its uncoupling, and thus the use of a staggered solution method where the field variables are solved alternatingly, avoids such difficulties and results in a stable, but sub-optimally converging solution method. In the present work, we evaluate the staggered against the monolithic solution approach and assess their computational performance in the analysis of fiber network failure. We then propose a hybrid solution technique that optimizes the performance and robustness of the computational analysis. It represents a matrix regularization technique that retains a positive definite element stiffness matrix while approaching the tangent stiffness matrix of the monolithic problem. Given the problems investigated in this work, the hybrid solution approach is up to 30 times faster than the staggered approach, where its superiority is most pronounced at large loading increments. The approach is general and may also accelerate the computational analysis of other failure problems.

Development of MoNbVTax refractory high entropy alloy with high strength at elevated temperature

Due to a monolithic body-centered cubic solid solution structure, refractory high-entropy alloys (HEA) always exhibit high strength and good softening resistance at both ambient and elevated temperatures. However, high density and low plasticity at ambient temperature limit their widespread applications as one kind of potential structural materials. The present study investigated the microstructure and mechanical properties of MoNbVTax HEA with various Ta conent at both ambient and elevated temperatures. The sample of MoNbVTax (x = 0, 0.3, 0.5, 0.7, 1) all shows a monolithic BCC solid solution phase with a dendritic microstructure. The micro-segregation of constituent elements was caused by the non-equilibrium solidification. The alloys exhibit superior high temperature performance at 1000 °C and 1200 °C, with a certain degree of plasticity at ambient temperature. The MoNbVTa0.5 alloy possesses both excellent high temperature performance (σ0.2 =697 MPa, εp = 35.4% at 1200 °C) and relatively low density (∼9.65 g/cm3), which is attributed to the solid solution strengthening effects and excellent thermal stability of constituent phase. Compared with commercial superalloys, the present MoNbVTa HEAs possess the advantages of high temperature performance, making it a potential engineering material for structural application.

A multicut generalized benders decomposition approach for the integration of process operations and dynamic optimization for continuous systems

The integration of process operations and dynamic optimization leads to large scale optimization problems whose monolithic solution is challenging. In this paper we propose a new formulation of the integrated planning, scheduling, and dynamic optimization problem for continuous single stage systems. We analyze the structure of the problem using Stochastic Blockmodeling and we show that the estimated structure can be used as the basis for a multicut Generalized Benders decomposition (GBD) algorithm, which can solve the problem in reduced computational time. Furthermore, we propose an accelerated hybrid multicut algorithm which can lead to further reduction in computational time. Through case studies, we analyze the computational performance of the proposed formulation and decomposition based solution algorithms.

Thermo-hygro-mechanically coupled modelling of wood including two-phase moisture diffusion for transient simulation of wooden structures at mechanical and climatic loads

A model to predict the thermo-hygro-mechanical behaviour of wood is introduced. The description of the transport processes of moisture and heat are combined with a model for the mechanical response. Moisture transport is represented by a two-phase multi-Fickian approach, considering bound water and water vapour. For the mechanical response, a moisture- and temperature-dependent, orthotropic, elastic material formulation is used. The theoretical basis of the model and the numerical implementation of the monolithic solution into a Finite Element framework are discussed as well as its verification and validation. With this model at hand, arbitrary wooden structures can be simulated in a transient manner subjected to climatic and mechanical loads. In the contribution, the approach is applied to the analysis of a panel painting by L. Cranach the Elder.

Consistent modeling of the coupling between crystallographic slip and martensitic phase transformation for mechanically induced loadings

AbstractA computational framework is proposed to model anisotropic metallic polycrystals subjected to mechanically induced loadings. It enables the simulation of crystal‐plasticity‐like phenomena at finite strains. The unified framework of thermodynamically consistent constitutive equations is formulated such that it couples the crystallographic slip and martensitic transformation theories. The constitutive description for the slip plasticity evolution incorporates an anisotropic hyperelastic law with self and latent‐hardening. The mechanically induced martensite formation kinematics is based on the crystallographic theory of martensitic transformations. The complete set of highly coupled equations is expressed in a single system of equations and solved by a monolithic solution procedure. It is based on the Newton–Raphson methodology and incorporates the complete linearisation leading to asymptotic quadratic rates of convergence. The quasi‐static discretized evolution equations are integrated with a fully implicit scheme, except for the critical resolved slip stresses, which employ the generalized midpoint rule. The plastic flow is integrated with an implicit exponential integrator to exactly preserve the plastic incompressibility. Viscous regularizations for both deformation mechanisms are pursued to overcome numerical difficulties and model the behavior over a wide range of strain‐rate sensitivities. Numerical examples are presented to demonstrate the efficiency and predictive capability of the methodology.

Evaluation of polymeric 3D printed adhesively bonded joints: effect of joint morphology and mechanical interlocking

PurposeThe purpose of this paper is to evaluate and exploit the combination of additive manufacturing polymeric technology and structural adhesives. The main advantage is to expand the maximum dimension of the 3D printed parts, which is typically limited, by joining the parts with structural adhesive, without losing strength and stiffness and keeping the major asset of polymeric 3 D printing: freedom of shape of the system and low cost of parts.Design/methodology/approachThe materials used in the paper are the following. The adhesive considered is a commercial inexpensive acrylic, quite similar to superglue, applicable with almost no surface preparation and fast curing, as time constraint is one of the key problems that affects industrial adhesive applications. The 3D printed parts were in acrylonitrile butadiene styrene (ABS), obtained with a Fortus 250mc FDM machine, from Stratasys. The work first compares flat overlap joint with joints designed to permit mechanical interlocking of the adherends and then to a monolithic component with the same geometry. Single lap, joggle lap and double lap joints are the configurations experimentally characterized following a design of experiment approach.FindingsThe results show a failure in the substrate, due to the low strength of the polymeric adherends for the first batch of typical bonded configurations, single lap, joggle lap and double lap. The central bonded area, with an increased global thickness, never does fail, and the adhesive is able to transfer the load both with and without mechanical interlocking. An additional set of scarf joints was also tested to promote adhesive failure as well as to retrieve the adhesive strength in this application. The results shows that bonding of polymeric AM parts is able to express its full potential compared with a monolithic solution even though the joint fails prematurely in the adherend due to the bending stresses and the notches present in the lap joints.Research limitations/implicationsBecause of the 3D printed polymeric material adopted, the results may be generalized only when the elastic properties of the adherends and of the adhesive are similar, so it is not possible to extend the findings of the work to metallic additive manufactured components.Practical implicationsThe paper shows that the adhesives are feasible way to expand the potentiality of 3 D printed equipment to obtain larger parts with equivalent mechanical properties. The paper also shows that the scarf joint, which fails in the adhesive first, can be used to extract information about the adhesive strength, useful for the designers which have to combine adhesive and additive manufactured polymeric parts.Originality/valueTo the best of the researchers’ knowledge, there are scarce quantitative information in technical literature about the performance of additive manufactured parts in combination with structural adhesives and this work provides an insight on this interesting subject. This manuscript provides a feasible way of using rapid prototyping techniques in combination with adhesive bonding to fully exploit the additive manufacturing capability and to create large and cost-effective 3 D printed parts.

A gradient-enhanced formulation for thermoviscoplastic metals accounting for ductile damage

In this work, we extend a gradient-enhancement formulation for ductile damage analysis to thermoviscoplastic metals. The original contribution of the study is the nonlocal extension of our previously developed thermoviscoplastic Gurson's model accounting for plastic work heat generation, thermal diffusion and void shearing mechanism. The gradient enrichment is applied to the porosity evolution equation by adding a nonlocal length scale parameter together with the Laplacian of the void volume fraction, following a void diffusion law. A mixed finite element formulation is used to discretize the system with an implicit time integration and a monolithic solution of the linearized system at every time step. Three ductile fracture problems involving plates under plane-strain conditions and tensile loading are simulated, in order to compare the local and nonlocal formulations. Results show that the nonlocal term delays the damage evolution and somewhat alleviates mesh sensitivity in terms of the stress-strain response and the porosity distribution along specific lines. Nevertheless, the most notable effect of the nonlocal formulation is the regularization (or smoothing) of the porosity field, reducing its gradients and mesh dependence, and leading to more consistent porosity contours. The choice of the nonlocal parameter is found to be dependent of the strain rate employed. • A gradient-enhanced thermoviscoplastic ductile damage formulation is proposed. • Coupling of Gurson's porous plasticity, void shearing mechanism, thermoviscoplasticity and thermal diffusion. • Three structural problems involving plates under plain strain assumption are numerically analyzed in detail. • The model is suitable to alleviate mesh dependence and to smooth the porosity field.

Efficient co‐solution of time step size and independent state in simulations of fluid‐driven fracture propagation with embedded meshes

AbstractWe present an efficient time‐continuation scheme for fluid‐driven fracture propagation problems in the extended finite element method framework. The approach applies a monolithic solution strategy to a fully coupled and implicit approximation of hydro‐mechanical systems in conjunction with simultaneous linear elastic propagation of multiple fractures. At the end of each time step, the process ensures that the weakest fracture tip is in an equilibrium propagation regime. Furthermore, the solution process provides an initialization procedure for the newly created fracture spaces and an a priori estimate of the stress intensity factor growth rate, improving simulation robustness, and efficiency. The solution process is validated using the Kristianovich‐Geertsma‐de Klerk analytical solution under the toughness‐ and viscosity‐dominated regimes. It is also extended to and demonstrated on problems with multiple fractures undergoing simultaneous propagation with stress shadow interactions. Numerical examples demonstrate that the solution process can reduce the required computational cost by one order of magnitude compared to other existing methods.

BarMan: A run-time management framework in the resource continuum

Over the last years, the number of IoT devices has grown exponentially, highlighting the current Cloud infrastructure limitations. In this regard, Fog and Edge computing began to move part of the computation closer to data sources by exploiting interconnected devices as part of a single heterogeneous and distributed system in a computing continuum viewpoint. Since these devices are typically heterogeneous in terms of performance, features, and capabilities, this perspective should encompass programming models and run-time management layers. This work presents and evaluates the BarMan open-source framework by implementing a Fog video surveillance use-case. BarMan leverages a task-based programming model combined with a run-time resource manager and the novel BeeR framework to deploy the application’s tasks transparently. This enables the possibility of considering aspects related to the energy and power dissipated by the devices and the single application. Moreover, we developed a task allocation policy to maximize application performance, considering run-time aspects, such as load and connectivity, of the time-varying available devices. Through an experimental evaluation performed on a real cluster equipped with heterogeneous embedded boards, we evaluated different execution scenarios to show the framework’s functionality and the benefit of a distributed approach, leading up to an improvement of 66% on the frame processing latency w.r.t. a monolithic solution.

A comparative study of fully implicit staggered and monolithic solution methods. Part I: Coupled bidomain equations of cardiac electrophysiology

This manuscript presents an extensive comparative analysis between a staggered and a monolithic solution technique for the bidomain equations of cardiac electrophysiology in the context of finite element method. For the monolithic solution technique, we adopt the work of Dal et al. (CMBBE 15: 645-656, 2012), where the parabolic and elliptic partial differential equations (PDEs) of the bidomain model are solved simultaneously and all rate equations are treated with an implicit time integration scheme. For the staggered scheme, however, we simply suggest a decoupled solution of the parabolic and elliptic PDE while keeping other aspects of the monolithic algorithm the same, such as utilization of the implicit time integration scheme, coupling of ordinary differential equations, which describe the state variables of cardiac electrophysiology, to the parabolic PDE and consideration of the transmembrane potential and the extracellular potential as degrees of freedom. Both solution algorithms are applied to several problems where we simulate regular planar wave propagations, scroll waves and externally applied electrical fields for different spatial and temporal resolution and material parameters. The comparison between the solution schemes is performed in terms of accuracy, efficiency and stability. We reveal that the suggested staggered solution scheme yields almost identical results with the monolithic scheme for physiological conductivity parameters and small time increments. Besides, the staggered scheme provides an enormous gain in computational efficiency, particularly, as the number of degrees of freedom is increased and we do not encounter any stability issue arising from the decoupled solution of the PDEs. Therefore, we conclude that the suggested fully implicit staggered solution algorithm has a tremendous application potential for the bidomain equations of cardiac electrophysiology.

Efficient monolithic solution of FE2 problems

AbstractThe concurrent FE2‐method is a very powerful and flexible computational tool for multi‐scale problems. However the computational effort is very high. The conventional, staggered (“nested Newton”) solution scheme solves the microscopic problem iteratively within each macroscopic Newton‐Raphson (NR) iteration, although the macroscopic deformation gradients as boundary conditions at the micro scale are only estimates. In this contribution a monolithic FE2scheme is proposed, solving the displacements of both scales in a common NR loop, which proved being faster by saving expansive micro‐scale iterations.

A generalised phase field model for fatigue crack growth in elastic–plastic solids with an efficient monolithic solver

We present a generalised phase field-based formulation for predicting fatigue crack growth in metals. The theoretical framework aims at covering a wide range of material behaviour. Different fatigue degradation functions are considered and their influence is benchmarked against experiments. The phase field constitutive theory accommodates the so-called AT1, AT2 and phase field-cohesive zone (PF-CZM) models. In regards to material deformation, both non-linear kinematic and isotropic hardening are considered, as well as the combination of the two. Moreover, a monolithic solution scheme based on quasi-Newton algorithms is presented and shown to significantly outperform staggered approaches. The potential of the computational framework is demonstrated by investigating several 2D and 3D boundary value problems of particular interest. Constitutive and numerical choices are compared and insight is gained into their differences and similarities. The framework enables predicting fatigue crack growth in arbitrary geometries and for materials exhibiting complex (cyclic) deformation and damage responses. The finite element code developed is made freely available at www.empaneda.com/codes.

Parallel multilevel domain decomposition preconditioners for monolithic solution of non-isothermal flow in reservoir simulation

In reservoir simulation, the non-isothermal flow adds a conservation of the energy equation with the involvement of a temperature variable to the complicated physics dynamics, which brings a higher nonlinearity of the corresponding PDE system and therefore exhibits significantly additional challenges on the linear preconditioner of the simulator. However, the commonly used block preconditioning techniques mostly focus on the isothermal petroleum model and do not have a specific treatment of the energy conservation equation for the non-isothermal flow. In this study, we propose and develop a family of multilevel restricted additive Schwarz (MRAS) preconditioners on parallel computers for the fully implicit solution of the non-isothermal flow in porous media. The proposed parallel reservoir simulator incorporates the restricted additive Schwarz preconditioner into a general multilevel preconditioning framework with the use of the domain decomposition technique and the multigrid method, which allows us to flexibly construct several types of multilevel Schwarz preconditioners by using the additive or multiplicative strategies. In particular, it follows the fully implicit discretization scheme for the monolithic solution, and therefore is unconditionally stable with the relaxation of the time step size by the stability condition. We numerically show that the proposed approach is highly robust and efficient for solving some non-isothermal flow problems with high nonlinearity, and good parallel scalabilities are obtained on a supercomputer with a large number of processors.

Integrated Electromagnetic-Thermal Approach to Simulate a GaN-Based Monolithic Half-Bridge for Automotive DC-DC Converter

New technological and packaging solutions are more and more being employed for power semiconductor switches in an automotive environment, especially the SiC- and GaN-based ones. In this framework, new front-end and back-end solutions have been developed, and many more are in the design stage. New and more integrated power devices are useful to guarantee the performances in electric vehicles, in terms of thermal management, size reduction, and low power losses. In this paper, a GaN-based system in package solution is simulated to assess the structure temperature submitted to a Joule heating power loss. The monolithic package solution involves a half-bridge topology, as well as a driver logic. A novel integrated electromagnetic and thermal method, based on finite element simulations, is proposed in this work. More specifically, dynamic electric power losses of the copper interconnections are computed in the first simulation stage, by an electromagnetic model. In the second stage, the obtained losses’ geometrical map is imported in the finite element thermal simulation, and it is considered as the input. Hence, the temperature distribution of the package’s copper traces is computed. The simulation model verifies the proper design of copper traces. The obtained temperature swing avoids any thermal-related reliability bottleneck.

Mixed variational formulations for structural topology optimization based on the phase-field approach

We propose a variational principle combining a phase-field functional for structural topology optimization with a mixed (three-field) Hu–Washizu functional, then including directly in the formulation equilibrium, constitutive, and compatibility equations. The resulting mixed variational functional is then specialized to derive a classical topology optimization formulation (where the amount of material to be distributed is an a priori assigned quantity acting as a global constraint for the problem) as well as a novel topology optimization formulation (where the amount of material to be distributed is minimized, hence with no pre-imposed constraint for the problem). Both formulations are numerically solved by implementing a mixed finite element scheme, with the second approach avoiding the introduction of a global constraint, hence respecting the convenient local nature of the finite element discretization. Furthermore, within the proposed approach it is possible to obtain guidelines for settings proper values of phase-field-related simulation parameters and, thanks to the combined phase-field and Hu–Washizu rationale, a monolithic algorithm solution scheme can be easily adopted. An insightful and extensive numerical investigation results in a detailed convergence study and a discussion on the obtained final designs. The numerical results clearly highlight differences between the two formulations as well as advantages related to the monolithic solution strategy; numerical investigations address both two-dimensional and three-dimensional applications.