Monolithic Solution Scheme Research Articles

A novel concurrent multiscale method based on the coupling of Direct FE2 and CPFEM

Performing concurrent simulations of macroscopic behaviors and microscopic structures using the crystal plasticity finite element method (CPFEM) presents a substantial difficulty with existing numerical techniques. To address this issue, a novel multi-scale method is proposed that couples CPFEM with a multiscale FEM, specifically Direct FE2. This facilitates the implementation of Direct CP-FE2 in this work. The micro representative volume elements (RVEs) equipped with a crystal plasticity constitutive model and the macro mesh are integrated into a monolithic solution scheme within the Direct FE2 framework. The proposed method integrates the multiscale simulation capability of Direct FE2 with the crystal plasticity model of CPFEM. Alpha titanium (α-Ti), which exhibits two distinct plastic mechanisms of slip and twinning, is chosen as the subject of investigation for conducting numerical experiments. The accuracy and efficiency of the Direct CP-FE2 model are evaluated through multiple plate tension and beam bending tests. The effective validation against the FEM model demonstrated the capability of Direct CP-FE2 to forecast macroscopic deformation behaviors. Meanwhile, the Direct CP-FE2 model can reveal the activation of slip/twinning systems and the evolution of crystal texture at a microscopic level. The influence of the grain orientation-dependent effect can be well considered into the macroscopic analysis with the help of Direct CP-FE2. Based on the testing examples, we demonstrate that the yield state of the macrostructure is enhanced when the crystal orientation is closer to the (0001) direction. Consequently, there exist very little crystal rotation behavior, hindering the evolution of the crystal texture.

Direct FE2 Based on Single-Integration Point for Modeling Damage Evolution of Materials with Micro-Cracks

The Direct FE2 method is a recently developed concurrent multi-scale simulation approach with a monolithic solution scheme, where the macroscopic and microscopic models are solved in a unit iteration process. The conventional Direct FE2 using quadrilateral element has some limitations in determining the failure state of a macro-element. In this work, we develop a new Direct FE2 method based on the Single-integration Point triangle element (Direct SP-FE2) and apply it to simulate the micro-crack-induced fracture problems for the first time. The cohesive element is inserted into a representative volume element (RVE), so that the failure process of a macro-element can be determined by the micro-crack propagation on RVE. Accordingly, the relationship between macro-deformations and microstructures can be established directly, thus avoiding complex derivation and utilization of a multi-scale damage constitutive model. In Direct SP-FE2, the failure state of a macro-element can be directly predicted by the RVE at the single-integration point inside, avoiding the uncertainty associated with predictions by multiple RVEs. As a result, compared to the conventional Direct FE2 with a quadrilateral element, the Direct SP-FE2 shows advantages in capturing complex geometric boundaries and predicting the damage evolution process and failure state more accurately. Various numerical examples are conducted to comprehensively validate the effectiveness of the present method in describing the influence of micro-cracks on macro-structure deformations. For the macro-damage behaviors, the simulation results by the Direct SP-FE2 show good agreement with the direct numerical simulation (DNS) results of the full FE model at a significantly lower computational time. With the developed Direct SP-FE2, a variety of complex micro-crack fracture problems in composite materials can be successfully modeled by manipulating the interior RVE structure.

Vectorized MATLAB Implementation of the Incremental Minimization Principle for Rate-Independent Dissipative Solids Using FEM: A Constitutive Model of Shape Memory Alloys

The incremental energy minimization principle provides a compact variational formulation for evolutionary boundary problems based on constitutive models of rate-independent dissipative solids. In this work, we develop and implement a versatile computational tool for the resolution of these problems via the finite element method (FEM). The implementation is coded in the MATLAB programming language and benefits from vector operations, allowing all local energy contributions to be evaluated over all degrees of freedom at once. The monolithic solution scheme combined with gradient-based optimization methods is applied to the inherently nonlinear, non-smooth convex minimization problem. An advanced constitutive model for shape memory alloys, which features a strongly coupled rate-independent dissipation function and several constraints on internal variables, is implemented as a benchmark example. Numerical simulations demonstrate the capabilities of the computational tool, which is suited for the rapid development and testing of advanced constitutive laws of rate-independent dissipative solids.

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.

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.

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.

A Unified Abaqus Implementation of the Phase Field Fracture Method Using Only a User Material Subroutine.

We present a simple and robust implementation of the phase field fracture method in Abaqus. Unlike previous works, only a user material (UMAT) subroutine is used. This is achieved by exploiting the analogy between the phase field balance equation and heat transfer, which avoids the need for a user element mesh and enables taking advantage of Abaqus’ in-built features. A unified theoretical framework and its implementation are presented, suitable for any arbitrary choice of crack density function and fracture driving force. Specifically, the framework is exemplified with the so-called AT1, AT2 and phase field-cohesive zone models (PF-CZM). Both staggered and monolithic solution schemes are handled. We demonstrate the potential and robustness of this new implementation by addressing several paradigmatic 2D and 3D boundary value problems. The numerical examples show how the current implementation can be used to reproduce numerical and experimental results from the literature, and efficiently capture advanced features such as complex crack trajectories, crack nucleation from arbitrary sites and contact problems. The code developed is made freely available.

Comparison of fully-coupled and sequential solution methodologies for enhanced geothermal systems

The simulation of enhanced geothermal systems (EGS) requires finding the solution to a highly coupled nonlinear set of partial differential equations. Verification of EGS modelling has been difficult due to a lack of analytical or semi-analytical solutions and is typically limited to a subset of processes. A comparison of alternative solution schemes is one method to numerically verify the coupling between all processes simultaneously and provide confidence in the solutions produced by the different solution schemes. This article presents the first such comparison of monolithic and sequential solution schemes for the modelling of EGS wells. A custom thermo-hydro-mechanical finite element model for an EGS well connected by planar fractures is developed along with two monolithic schemes; one using the Newton–Raphson iterative method and another using the Newton–Raphson iterative method modified by Aitken’s Δ2 relaxation method. The two monolithic schemes are compared against two sequential schemes and one loosely-coupled scheme in terms of accuracy and computational efficiency. The monolithic schemes are shown to have an optimal rate of convergence with respect to mesh and timestep refinement. The monolithic Newton–Raphson scheme is shown to require small timesteps that violate the minimum timestep size of the β-method of time integration, introducing spurious oscillations into the solution. Aitken’s Δ2 relaxation method is shown to improve the ability of the Newton–Raphson scheme to find a converged solution at larger timesteps, avoiding spurious oscillations and reducing overall computation time. The sequential solution schemes are shown to provide optimal rates of convergence with respect to mesh and timestep refinement and converge to the same solution as the monolithic solution schemes, verifying the results of both methodologies and previous modelling efforts. It is demonstrated that the sequential solution schemes are computationally more expensive than the monolithic schemes when seeking well-converged solutions. The loosely-coupled scheme demonstrates an optimal rate of convergence with mesh and timestep refinement, but is also shown to be much less accurate than the other schemes. The recommended solution scheme for the efficient simulation of EGSs is thus the Newton–Raphson scheme with Aitken’s Δ2 relaxation.

Phase field fracture modelling using quasi-Newton methods and a new adaptive step scheme

We investigate the potential of quasi-Newton methods in facilitating convergence of monolithic solution schemes for phase field fracture modelling. Several paradigmatic boundary value problems are addressed, spanning the fields of quasi-static fracture, fatigue damage and dynamic cracking. The finite element results obtained reveal the robustness of quasi-Newton monolithic schemes, with convergence readily attained under both stable and unstable cracking conditions. Moreover, since the solution method is unconditionally stable, very significant computational gains are observed relative to the widely used staggered solution schemes. In addition, a new adaptive time increment scheme is presented to further reduce the computational cost while allowing to accurately resolve sudden changes in material behavior, such as unstable crack growth. Computation times can be reduced by several orders of magnitude, with the number of load increments required by the corresponding staggered solution being up to 3000 times higher. Quasi-Newton monolithic solution schemes can be a key enabler for large scale phase field fracture simulations. Implications are particularly relevant for the emerging field of phase field fatigue, as results show that staggered cycle-by-cycle calculations are prohibitive in mid or high cycle fatigue. The finite element codes are available to download from www.empaneda.com/codes.

A computationally efficient locking free numerical framework for modeling visco-hyperelastic dielectric elastomers

In this paper, we present a finite element formulation for simulating the electromechanical behavior of nonlinear visco-hyperelastic dielectric elastomers undergoing isochoric finite strain deformations. The incompressible nature of dielectric elastomer results in the problem of volumetric locking. In order to alleviate this problem various techniques, such as selective reduce integration method, F-bar method, and mixed formulation have been used. In the present numerical framework, we treat this problem by adapting the computationally efficient J-bar method. This method of alleviating locking is based on the assumption of the dependence of volumetric density function only on the smoothed volumetric part of the multiplicatively decomposed deformation gradient. A staggered solution algorithm is used for solving the coupled nonlinear equations by decoupling displacement and electric potential fields. The present formulation is implemented into an in-house finite element program. Firstly, the finite element implementation of the proposed formulation is validated and demonstrated by investigating the effect of viscosity parameter on the electromechanical phenomenon of homogeneously deforming dielectric elastomers, i.e., voltage driven creep and pull-in instability. Further, for an in-homogeneously deforming bi-layered bending actuator, the computational efficiency of the proposed J-bar method employed in conjunction with the staggered solution approach is established over the conventional F-bar and multi-field formulations used with the monolithic solution scheme. Finally, we utilize the proposed formulation for investigating the effect of visco-elasticity on the electromechanical response of complex dielectric elastomeric actuators involving inhomogeneous deformations such as, bilayer bending actuators, grippers, and buckling pumps, when subjected to monotonic and constant electrical loads.

A computational approach for thermo-elasto-plastic frictional contact based on a monolithic formulation using non-smooth nonlinear complementarity functions

A new monolithic solution scheme for thermo-elasto-plasticity and thermo-elasto-plastic frictional contact with finite deformations and finite strains is presented. A key feature is the reformulation of all involved inequality constraints, namely those of Hill’s orthotropic yield criterion as well as the normal and tangential contact constraints, in terms of non-smooth nonlinear complementarity functions. Using a consistent linearization, this system of equations can be solved with a non-smooth variant of Newton’s method. A quadrature point-wise decoupled plastic constraint enforcement and the use of so-called dual basis functions in the mortar contact formulation allow for a condensation of all additionally introduced variables, thus resulting in an efficient formulation that contains discrete displacement and temperature degrees of freedom only, while, at the same time, an exact constraint enforcement is assured. Numerical examples from thermo-plasticity, thermo-elastic frictional contact and thermo-elasto-plastic frictional contact demonstrate the wide range of applications covered by the presented method.

Phase-Field Modelling of Crack Propagation in Anisotropic Polycrystalline Materials

Nowadays, products consisting of polycrystalline materials have been widely used in engineering applications, e.g. automobile and renewable energy. The macroscopic defects are generally strongly influenced by the fracture behavior of the polycrystalline materials at the meso- and microscopic level. In this paper, the proposed phase-field model for anisotropic fracture, which accounts for the preferential cleavage directions within each randomly oriented crystal, as well as an anisotropic material behavior with cubic symmetries, has been used to simulate the complex crack pattern in solar-grade polycrystalline silicon in a robust and straightforward manner. Furthermore, multi-field coupled finite element problems are performed with monolithic solution schemes. A representative numerical example for crack propagation in polycrystals is carried out. Finally, a summary of the numerical results in polycrystalline materials is presented and an outlook for next work steps is given.

Phase-Field Modelling of Crack Propagation in Anisotropic Polycrystalline Materials

Abstract Nowadays, products consisting of polycrystalline materials have been widely used in engineering applications, e.g. automobile and renewable energy. The macroscopic defects are generally strongly influenced by the fracture behavior of the polycrystalline materials at the meso- and microscopic level. In this paper, the proposed phase-field model for anisotropic fracture, which accounts for the preferential cleavage directions within each randomly oriented crystal, as well as an anisotropic material behavior with cubic symmetries, has been used to simulate the complex crack pattern in solar-grade polycrystalline silicon in a robust and straightforward manner. Furthermore, multi-field coupled finite element problems are performed with monolithic solution schemes. A representative numerical example for crack propagation in polycrystals is carried out. Finally, a summary of the numerical results in polycrystalline materials is presented and an outlook for next work steps is given.

A Monolithic Solution Scheme for a Phase Field Model of Ductile Fracture

AbstractA phase field model for elastic‐plastic fracture is presented, which is based on an energy functional composed of an elastic energy contribution, a plastic dissipation potential and a fracture energy. The coupling of the mechanical fields with the fracture field is modeled by a degradation function. Due to the proposed coupling it is possible to solve the global system of differential equations in a monolithic iterative solution scheme. Numerical simulations are presented, where the choice of the degradation function is investigated and a staggered is compared to monolithic iteration scheme. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A feedback-loop extended stress fiber growth model with focal adhesion formation

Contractile cells play a prominent role in the adaptive nature of biological tissues. Contractility is mainly attributed to the growth of the tension dependent actomyosin bundles called stress fibers within the cytoskeleton. Stress fibers extend along the length of the cell and end at focal adhesions on the cell membrane. At the focal adhesion junctions on the cell membrane the integrin proteins are capable of sensing the environment, thereby making the cellular behavior dependent on the cell supporting substrate. It has been observed that the growth of stress fibers influences focal adhesions and vice-versa, resulting in a continuous cross-talk between different processes in the cell. Recent experiments have shown that cells subjected to uni-axial cyclic loading, depending on the substrate properties reorient themselves in a direction away from the loading direction, exhibiting strain avoidance.Mathematical models are important to understand the dependence of the cellular behavior on the substrate properties along with feedback mechanisms and are further used in designing in-vitro experiments. The coupling of the models for stress fibers and focal adhesions results in a non-linear bio-chemo-mechanical problem. In this contribution, we present the positive influence of the growth of focal adhesions along with a mechanosensitive feedback loop on the stress fiber growth and further reveal the characteristics of the re-orientation process due to cyclic loading. We use a non-linear Hill-type model to capture the growth of the active stress involved in the evolution law for the stress fibers and a thermodynamical approach to model the focal adhesions. A highly stable and reliable monolithic solution scheme is used to solve the governing system of coupled equations. Finally, we validate our simulation results with experimental results in regard to different loading conditions.

On phase field modeling of ductile fracture

AbstractRecently, different extensions of phase field fracture models to ductile fracture scenarios where the crack evolution is affected by plastic deformation have been proposed. In most of these phase field models for ductile fracture, the coupling of the elastic‐plastic material laws and the phase field variable is rather complicated and staggered finite element solution schemes are employed in order to avoid the computation of the complicated coupling terms in the tangent matrix. In this work, we propose an elastic‐plastic phase field fracture model, where the coupling is such that a monolithic solution is possible. Through the proposed coupling of the elastic‐plastic material law and the phase field order parameter, a reinterpretation of the meaning of the plastic material parameters is necessary in order to obtain reasonable effective material properties for the ductile phase field fracture model that can be related to measured data. The respective relations are derived from an analysis of the model in one dimension. Finite element simulations of plane strain fracture problems demonstrate the appropriateness of the derived effective material parameters and the applicability of the monolithic solution scheme. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Volumetric coupling approaches for multiphysics simulations on non‐matching meshes

SummaryIn finite element analysis of volume coupled multiphysics, different meshes for the involved physical fields are often highly desirable in terms of solution accuracy and computational costs. We present a general methodology for volumetric coupling of different meshes within a monolithic solution scheme. A straightforward collocation approach is compared to a mortar‐based method for nodal information transfer. For the latter, dual shape functions based on the biorthogonality concept are used to build the projection matrices, thus further reducing the evaluation costs. We give a detailed explanation of the integration scheme and the construction of dual shape functions for general first‐order and second‐order Langrangian finite elements within the mortar method, as well as an analysis of the conservation properties of the projection operators. Moreover, possible incompatibilities due to different geometric approximations of curved boundaries are discussed. Numerical examples demonstrate the flexibility of the presented mortar approach for arbitrary finite element combinations in two and three dimensions and its applicability to different multiphysics coupling scenarios. Copyright © 2016 John Wiley & Sons, Ltd.

A line search assisted monolithic approach for phase-field computing of brittle fracture

Phase-field modeling of fracture phenomena in solids is a very promising approach which has gained popularity within the last decade. However, within the finite element framework, already a two-dimensional quasi-static phase-field formulation is computationally quite demanding, mainly for the following reasons: (i) the need to resolve the small length scale inherent to the diffusive crack approximation calls for extremely fine meshes, at least locally in the crack phase-field transition zone, (ii) due to non-convexity of the related free-energy functional, a robust, but slowly converging staggered solution scheme based on algorithmic decoupling is typically used. In this contribution we tackle problem (ii) and propose a faster and equally accurate approach for quasi-static phase-field computing of (brittle) fracture using a monolithic solution scheme which is accompanied by a novel line search procedure to overcome the iterative convergence issues of non-convex minimization. We present a detailed critical evaluation of the approach and its comparison with the staggered scheme in terms of computational cost, accuracy and robustness.

A Pian–Sumihara type element for modeling shear bands at finite deformation

A monolithic numerical solution of a partial differential equation (PDE) model for shear bands, which includes a thermal softening rate dependent plastic flow rule and finite thermal conductivity, is presented. The formulation accounts for large deformation kinematics and includes incrementally objective treatment of the hypoplastic constitutive relations. Regularization is achieved by including finite thermal conductivity, which informs the PDE system of a length scale, governed by competition between shear heating and thermal diffusion. The monolithic solution scheme is then used to eliminate splitting errors during the solution of the discretized system. The scheme is presented in a general, mixed formulation, which allows for many choices of shape functions. We study and compare two elements, which have been implemented with the monolithic nonlinear solver: the Irreducible Shear Band Quad (ISBQ) and the Pian Sumihara Shear Band Quad (PSSBQ). ISBQ employs the same interpolation as an irreducible four node quad while PSSBQ is a mixed, assumed stress element. The algorithmic approximations to the Lie derivative and Jaumann rate of Kirchhoff stress are available in the literature for ISBQ type elements, and are derived in this paper for the PSSBQ. These expressions are used to achieve an incrementally objective formulation. It is found that the PSSBQ converges faster than the ISBQ with mesh refinement, and that the convergence of the ISBQ can be improved with a remeshing procedure.