Iterative Coupling Scheme Research Articles

Phase-field modeling and effective simulation of non-isothermal reactive transport

We consider single-phase flow with solute transport where ions in the fluid can precipitate and form a mineral, and where the mineral can dissolve and release solute into the fluid. Such a setting includes an evolving interface between fluid and mineral. We approximate the evolving interface with a diffuse interface, which is modeled with an Allen–Cahn equation. We also include effects from temperature such that the reaction rate can depend on temperature, and allow heat conduction through fluid and mineral. As Allen–Cahn is generally not conservative due to curvature-driven motion, we include a reformulation that is conservative. This reformulation includes a non-local term that makes the use of standard Newton iterations for solving the resulting non-linear system of equations very slow. We instead apply L-scheme iterations, which can be proven to converge for any starting guess, although giving only linear convergence. The three coupled equations for diffuse interface, solute transport and heat transport are solved via an iterative coupling scheme. This allows the three equations to be solved more efficiently compared to a monolithic scheme, and only few iterations are needed for high accuracy. Through numerical experiments we highlight the usefulness and efficiency of the suggested numerical scheme and the applicability of the resulting model.

Modular Framework for the Solution of Boundary-coupled Multiphysics Problems

This paper presents a modular multiphysics framework developed for OpenFOAM. The framework is built around an iterative implicit coupling scheme based on a multi-region partitioned approach. This scheme allows the implementation of formal implicit time-marching schemes, which improves the stability of strongly interacting coupled problems. This methodology allows physical interactions to be handled through specifically designed interface boundary conditions. It also allows region-specific solvers to be implemented as modular class solvers. The coupling methodology is handled with a main program that manages solver-specific actions. The aim of this framework is to facilitate the implementation and testing of new multiphysics coupling problems in an integrated code structure. To show the capabilities of the framework to integrate new physics, solvers and boundary conditions requirements are discussed. Also, three validated examples involving fluid-structure interactions, conjugate heat transfer, and fluid-structure-thermal interactions are presented. Although all these problems are boundary-coupled multiphysics problems, the framework is conceptually not limited to this kind of problems. The benefit of this work to the OpenFOAM community is a general and modular framework that facilitates the setup and solution of diversified multiphysics problems, and that illustrates the implementation of modular interface boundary conditions between physics regions.

Convergence analysis of single rate and multirate fixed stress split iterative coupling schemes in heterogeneous poroelastic media

AbstractRecently, the accurate modeling of flow‐structure interactions has gained more attention and importance for both petroleum and environmental engineering applications. Of particular interest is the coupling between subsurface flow and reservoir geomechanics. Different single rate and multirate iterative and explicit coupling schemes have been proposed and analyzed in the past. In addition, Banach fixed point contraction results were obtained for iterative coupling schemes, and conditionally stable results were obtained for explicit coupling schemes. In this work, we will consider the mathematical analysis of the single rate and multirate fixed stress split iterative coupling schemes for spatially heterogeneous poroelastic media. We will re‐establish the contractivity for both schemes in the localized case, and we will show that heterogeneities come at the expense of imposing more restricted conditions on the number of fine flow time steps that can be taken within one coarse mechanics time step in the multirate case. Our mathematical analysis is supplemented by numerical simulations validating our derived upper bounds. To the best of our knowledge, this is the first rigorous mathematical analysis of the multirate fixed‐stress split iterative coupling scheme in heterogeneous poroelastic media.

A CFD-FEM based partitioned fluid structure interaction model to investigate surface cracks in elastohydrodynamic lubricated line contacts

This paper presents a partitioned, two-dimensional fluid structure interaction (FSI) model developed to investigate the effects of surface cracks in rolling/sliding line contacts operating under elastohydrodynamic lubrication (EHL). The lubricant behavior as it flows over and through the crack is determined by solving the Navier-Stokes equations using the Ansys Fluent computational fluid dynamics (CFD) solver. The structural response of the solid body is governed via an elastic-plastic constitutive relationship and solved using the Ansys Mechanical finite element solver. The exchange of (i) forces due to fluid pressure, and (ii) displacements due to solid deformation across the fluid-solid interface (including the crack faces) is facilitated by an iterative implicit coupling scheme. A smoothing based dynamic meshing technique is utilized to capture the deformation of the fluid region. The FSI model developed for this investigation is capable of modeling surface cracks with inclined geometries, overcoming the limitations of the classical Reynolds based approach. The fluid pressure results are presented for different loads, speeds and slide-to-roll ratios. The effects of crack geometry (i.e. location, dimension, inclination, crack tip radius, etc.) on fluid pressure and structural response are investigated. The FSI model presents the details of pressure distribution, fluid streamlines and stress concentration in the solid simultaneously. The model results can be used to provide accurate boundary conditions on the crack faces for crack propagation modeling in rolling contact fatigue. The results of this investigation identify the crack geometries that affect fatigue life of rolling elements in EHL contact by predicting the location and severity of stress concentrations in the material.

A new quasi-3-D model with a dual iterative coupling scheme for simulating unsaturated-saturated water flow and solute transport at a regional scale

Soil water in an unsaturated zone is the major source of groundwater recharge, and likewise soil contaminants are the major source of groundwater contamination. It is thus essential to jointly consider soil water movement and solute transport together with groundwater movement and solute transport, which is particularly important when addressing regional-scale problems. This study developed a new quasi-3-D model for simulating regional-scale saturated–unsaturated water movement and solute transport in arid and semi-arid agricultural areas. The unsaturated soil water movement and solute transport processes are simulated by a 1-D soil water and solute transport model referred to as UBMOD, and the saturated groundwater movement and solute transport are simulated by MODFLOW and MT3DMS. In the quasi-3-D model, the soil water movement and groundwater movement modules are iteratively coupled and the soil solute transport and groundwater solute transport modules are iteratively couple. This is referred to as a dual iterative coupling scheme. Strength and weakness of the quasi-3-D model and the coupling scheme were evaluated with two synthetic cases and a real-world application. Results indicate satisfactory performance of the model, which demonstrates the reliability of the dual iterative coupling scheme. Besides, the impacts of spatial and temporal discretization and stress period on model stability and accuracy has been evaluated. And the model is more computationally efficient than the fully numerical methods based on Richards’ equation when the problem is a 2D/3D problem with a scale larger than 10 m. Moreover, the application to the Hetao Irrigation District indicates that the developed model is effective and efficient for the real-world regional-scale agricultural problems. Above all, it is concluded that the new model is more suitable than existing models for simulating regional-scale unsaturated–saturated soil water movement and solute transport in arid and semi-arid agricultural areas.

A novel smoothed particle hydrodynamics and finite element coupling scheme for fluid–structure interaction: The sliding boundary particle approach

A novel numerical formulation for solving fluid–structure interaction (FSI) problems is proposed where the fluid field is spatially discretized using smoothed particle hydrodynamics (SPH) and the structural field using the finite element method (FEM). As compared to fully mesh- or grid-based FSI frameworks, due to the Lagrangian nature of SPH this framework can be easily extended to account for more complex fluids consisting of multiple phases and dynamic phase transitions. Moreover, this approach facilitates the handling of large deformations of the fluid domain respectively the fluid–structure interface without additional methodological and computational efforts. In particular, to achieve an accurate representation of interaction forces between fluid particles and structural elements also for strongly curved interface geometries, the novel sliding boundary particle approach is proposed to ensure full support of SPH particles close to the interface. The coupling of the fluid and the structural field is based on a Dirichlet–Neumann partitioned approach, where the fluid field is the Dirichlet partition with prescribed interface displacements and the structural field is the Neumann partition subject to interface forces. To overcome instabilities inherent to weakly coupled schemes an iterative fixed-point coupling scheme is employed. Several numerical examples in form of well-known benchmark tests are considered to validate the accuracy, stability, and robustness of the proposed formulation. Finally, the filling process of a highly flexible thin-walled balloon-like container is studied, representing a model problem close to potential application scenarios of the proposed scheme in the field of biomechanics.

Coupling 3D geomechanics to classical sedimentary basin modeling: From gravitational compaction to tectonics

Classical sedimentary basin simulators account for simplified geomechanical models that describe material compaction by means of phenomenological laws relating porosity to vertical effective stress. In order to overcome this limitation and to deal with a comprehensive poromechanical framework, an iterative coupling scheme between a basin modeling code and a mechanical finite element code is adopted. This work focuses on the porous material constitutive law specifically devised to couple 3D geomechanics to basin modeling. The sediment material is considered as an isotropic fully saturated poro-elastoplastic medium undergoing large irreversible strains. Special attention is given to the development of a hardening law capable of reproducing the same porosity evolution as provided by the standard basin simulator when the sediment material is submitted to gravitational compaction under oedometric conditions. A synthetic case is used to illustrate the ability of the proposed workflow to integrate horizontal deformations in the basin model as such effects cannot be captured by the simplified geomechanics of the standard basin code. The results obtained by the coupled simulation demonstrate that horizontal compression may significantly contribute to overpressure development and brittle failure of the basin seal rocks, highlighting the importance of a coupled approach to simulate complex tectonic history.

A Schwarz iterative method to evaluate ocean–atmosphere coupling schemes: implementation and diagnostics in IPSL-CM6-SW-VLR

Abstract. State-of-the-art Earth system models, like the ones used in the Coupled Model Intercomparison Project Phase 6 (CMIP6), suffer from temporal inconsistencies at the ocean–atmosphere interface. Indeed, the coupling algorithms generally implemented in those models do not allow for a correct phasing between the ocean and the atmosphere and hence between their diurnal cycles. A possibility to remove these temporal inconsistencies is to use an iterative coupling algorithm based on the Schwarz iterative method. Despite its large computational cost compared to standard coupling methods, which makes the algorithm implementation impractical for production runs, the Schwarz method is useful to evaluate some of the errors made in state-of-the-art ocean–atmosphere coupled models (e.g., in the representation of the processes related to diurnal cycle), as illustrated by the present study. IPSL-CM6-SW-VLR is a low-resolution version of the IPSL-CM6 coupled model with a simplified land surface model, implementing a Schwarz iterative coupling scheme. Comparisons between coupled solutions obtained with this new scheme and the standard IPSL coupling scheme (referred to as the parallel algorithm) show large differences after sunrise and before sunset, when the external forcing (insolation at the top of the atmosphere) has the fastest pace of change. At these times of the day, the difference between the two numerical solutions is often larger than 100 % of the solution, even with a small coupling period, thus suggesting that significant errors are potentially made with current coupling methods. Most of those differences can be strongly reduced by making only two iterations of the Schwarz method, which leads to a doubling of the computing cost. Besides the parallel algorithm used in IPSL-CM6, we also test a so-called sequential atmosphere-first algorithm, which is used in some coupled ocean–atmosphere models. We show that the sequential algorithm improves the numerical results compared to the parallel one at the expanse of a loss of parallelism. The present study focuses on the ocean–atmosphere interface with no sea ice. The problem with three components (ocean–sea ice–atmosphere) remains to be investigated.

An optimally-coupled multi-time stepping method for transient heat conduction simulation for additive manufacturing

With growing application of additive manufacturing (AM) in industry, simulation of transient heat conduction during the AM process has drawn increasing attention. However, detailed simulation with acceptable accuracy is extremely time consuming because of the large range of time scales involved in the problem. In this work, a multi-time stepping method using different timesteps in multiple subdomains is developed to accelerate transient heat conduction simulation. A Dirichlet–Robin iterative coupling scheme is proposed to enforce continuity at the interfaces between different subdomains. This allows numerical stability of the multi-timestep​ model, even with disparate materials or meshes on either side of the subdomain interfaces. Specifically, an optimal approximation for the Schur complement matrix is derived using a one-dimensional (1D) model, which avoids the instabilities encountered when using a Dirichlet–Neumann coupling formulation and allows convergence of the iterative coupling scheme in only one iteration at each timestep. Several numerical examples are studied to evaluate the stability, convergence and performance of the method. It is found that the proposed method is robust and stable for all choices of material parameters, unlike the conventional Dirichlet–Neumann coupling scheme. To demonstrate the use of the method for AM, a simulation representative of a single laser track is performed and compared with line heat source model approach. It is found that the proposed method not only accelerates the simulation by a factor of more than 100, but gives very good accuracy for temperature in the melt pool region.

Fluid–Structure Interaction Modeling of Elastohydrodynamically Lubricated Line Contacts

Abstract This paper presents a partitioned fluid–structure interaction (FSI) solver to model elastohydrodynamic lubrication (EHL) of line contacts. The FSI model was constructed using the multiphysics simulation software ansys, wherein an iterative implicit coupling scheme is implemented to facilitate the interaction between fluid and solid components. The model uses a finite volume method (FVM) based computational fluid dynamics (CFD) solver to determine the lubricant flow behavior using the Navier–Stokes equations. Additionally, the finite element method (FEM) is utilized to model the structural response of the solid. Fluid cavitation, compressibility, non-Newtonian lubricant rheology, load balance algorithm, and dynamic meshing were incorporated in the FSI model. The pressure and film thickness results obtained from the model are presented for a wide range of loads, speeds, slide to roll ratios (SRR), surface dent, material properties (elastic plastic), etc. The model presents a detailed understanding of EHL contacts by removing any assumptions relative to the Reynolds equation. It provides the (i) two-dimensional variation of pressure, viscosity, etc., in the fluid and (ii) stress, elastic/plastic strain in the solid, simultaneously. The FSI model is robust, easy to implement, and computationally efficient. It provides an effective approach to solve sophisticated EHL problems. The FSI model was used to investigate the effects of surface dents, plasticity and material inclusions under heavily loaded lubricated line contacts as can be found in gears and rolling element bearings. The results from the model exhibit excellent corroboration with published results based on the Reynolds equation solvers.

A novel iterative method based on fixed stress rates for hydromechanical problems with nonlinear constitutive relationships

This paper focuses on the development of alternative coupling strategies for hydromechanical applications considering highly nonlinear constitutive models to represent the behavior of porous media. We present a novel consistent iterative coupling formulation based on fixed stress rates to define the coupling terms between mass balance and equilibrium equations. We show that controlling stress rates allows the application of an adaptive iterative coupling scheme with single or multi-rate solutions. The proposed methodology has been implemented in a framework that manages sequential simulations, exchanges of information between models, and checks the convergence of the iterative process. We present several numerical tests in which we evaluate the stability, accuracy, and performance of the proposed coupling technique. From the results, we confirm that the rate form of the fixed stress split guarantees solution stability and yields results as accurate as those obtained through fully implicit solutions, even for highly nonlinear constitutive behavior.

A tight coupling scheme for smooth/non-smooth multibody co-simulation of a particle damper

To simulate coupled problems, composed of smooth and non-smooth particle subsystems, a tight, i.e. iterative coupling scheme is implemented. In the proposed setup, the non-smooth solver iterations are enclosed in each iteration of the smooth solver at every time step. The stability properties of the coupling scheme are tested using Dahlquist’s test equations for co-simulation methods, highlighting the higher degree of stability of the tight coupling scheme, compared to the loose, non-iterative one. Results also denote that the properties of the solvers used to analyse the subsystems play an important role in the stability of co-simulation. The performance of the proposed scheme is further assessed by two illustrative examples: a particle damper subjected to prescribed motion, and a cantilever beam, modelled using finite elements, with a particle damper mounted at the free end. The proposed numerical solutions agree well with results from reduced order models and experiments, which demonstrates that the proposed scheme can effectively handle coupled problems of multibody systems and particle dampers.

Comprehensive Investigation of Loss Calculation and Sequential Iterative Fluid-Solid Coupling Schemes for High-Speed Switched Reluctance Motors

This article proposes a comprehensive investigation method for loss calculation and temperature rise prediction in high-speed and high-power switched reluctance machines (SRMs). The losses are analyzed in details and the sequential iterative fluid-solid coupling (SIFSC) scheme is proposed for thermal analysis. In the iterative design stage of the SRM, it is important and urgent to figure out the power losses and temperature distribution in the motor, which affects the endurance of the motor operation, such as the maximum speed and highest operating temperature. A simplified finite element (FE) model of a turn and a proximity loss separation method are developed for the skin and proximity (SP) loss reduction. Then, an additional eddy current loss calculation method is developed with a simplified FE model of the winding. Additionally, the SIFSC scheme is further proposed and implemented by sequentially calculating the fluid and solid fields for airflow velocity and temperature distribution. Compared to conventional thermal calculation schemes, the proposed SIFSC scheme is faster than direct coupling methods and more accurate than thermal network methods. The accuracy and effectiveness of this scheme is verified by the simulation and experimental test for a 160 kW and 18000 r/min 6/2 SRM prototype.

Coupled numerical approach combining X-FEM and the embedded discrete fracture method for the fluid-driven fracture propagation process in porous media

A coupled simulation strategy combining the embedded discrete fracture method (EDFM) and the extended finite element method (X-FEM) is developed to simulate the fluid driven fracture propagation process in porous media. This physical process includes three strong coupling mechanics: fluid flow in fractures and porous media, solid deformation with fractures, and fracture propagations. The EDFM and X-FEM are used to simulate fracture-related fluid mechanics and solid mechanics, respectively, with information exchanged under the iterative numerical coupling scheme. Mathematical equations on how to link these independent modules as well as numerical techniques on how to accelerate the coupling convergence rate are discussed in detail.Both X-FEM and EDFM avoid the cumbersome construction of unstructured grids to capture fracture paths and also avoid the remeshing for the fracture growth. They are first validated via benchmark problems individually and then are coupled to simulate fracture propagation problems in two dimensions and in three dimensions. Simulated multiphysics fields meet understandings qualitatively, and simulated fracture parameters (length, width and net pressure) match with analytical solutions quantitatively.

Partitioned Strong Coupling of Discrete Elements with Large Deformation Structural Finite Elements to Model Impact on Highly Flexible Tension Structures

This article presents a staggered approach to couple the interaction of very flexible tension structures with large deformations, described with the finite element method (FEM), and objects undergoing large, complex, and arbitrary motions discretized with particle methods, in this case the discrete element method (DEM). The quantitative solution quality and convergence rate of this partitioned approach is highly time step dependent. Thus, the strong coupling approach is presented here, where the convergence is achieved in an iterative manner within each time step. This approach helps increase the time step size significantly, decreases the overall computational costs, and improves the numerical stability. Moreover, the proposed algorithm enables the application of two independent, standalone codes for simulating DEM and structural FEM as blackbox solvers. Systematic evaluations of the newly proposed iterative coupling scheme with respect to accuracy, robustness, and efficiency as well as cross comparisons between strong and weak FEM‐DEM coupling approaches are performed. Additionally, the approach is validated against the rest position of an impacting object, and further examples with objects impacting highly flexible protection structures are presented. Here, the protection nets are described with nonlinear structural finite elements and the impacting objects as DEM elements. To allow the interested reader to independently reproduce the results, detailed code and algorithm descriptions are included in the appendix.

Analysis of dynamic ruptures generating seismic waves in a self-gravitating planet: An iterative coupling scheme and well-posedness

We study the solution of the system of equations describing the dynamical evolution of spontaneous ruptures generated in a prestressed elastic-gravitational deforming body and governed by rate and state friction laws. We propose an iterative coupling scheme based on a weak formulation with nonlinear interior boundary conditions, both for continuous time and with implicit discretization (backward Euler) in time. We regularize the problem by introducing viscosity. This guarantees the convergence of the scheme for solutions of the regularized problems in both cases. We also make precise the conditions on the relevant coefficients for convergence to hold.

A multi-rate iterative coupling scheme for simulating dynamic ruptures and seismic waves generation in the prestressed earth

We present a novel method to simulate the dynamic evolution of spontaneous ruptures governed by rate- and state-dependent friction laws and the interaction with seismic waves in a prestressed elastically deforming body. We propose a multi-rate iterative coupling scheme based on the variational form of the elastic-gravitational equations, and discretize employing a discontinuous Galerkin method, with nonlinear interior boundary conditions being weakly imposed across the fault surface as numerical fluxes. We introduce necessary interface jump penalty terms as well as an artificial viscous regularization, with the conditions for penalty and viscosity coefficients given based on an energy estimate and a convergence analysis. In the multi-rate scheme, an implicit-explicit Euler scheme in time is invoked, and the time step for the evolution of friction is chosen significantly finer than that for wave propagation and scattering. This is facilitated by the iterative scheme through the underlying decoupling where the linear, elastic wave equation plays the role of a Schur-complement to the friction model. A nonlinearly constrained optimization problem localized to each element on the rupture surface is then formulated and solved using the Gauss-Newton method. We test our algorithm on several benchmark examples and illustrate the generality of our method for realistic rupture simulations.

Convergence of the undrained split iterative scheme for coupling flow with geomechanics in heterogeneous poroelastic media

Recently, an accurate coupling between subsurface flow and reservoir geomechanics has received more attention in both academia and industry. This stems from the fact that incorporating a geomechanics model into upstream flow simulation is critical for accurately predicting wellbore instabilities and hydraulic fracturing processes. One of the recently introduced iterative coupling algorithms to couple flow with geomechanics is the undrained split iterative coupling algorithm as reported by Kumar et al. (2016) and Mikelic and Wheeler (Comput. Geosci. 17: 455–461 2013). The convergence of this scheme is established in Mikelic and Wheeler (Comput. Geosci. 17:455–461 2013) for the single rate iterative coupling algorithm and in Kumar et al. (2016) for the multirate iterative coupling algorithm, in which the flow takes multiple finer time steps within one coarse mechanics time step. All previously established results study the convergence of the scheme in homogeneous poroelastic media. In this work, following the approach in Almani et al. (2017), we extend these results to the case of heterogeneous poroelastic media, in which each grid cell is associated with its own set of flow and mechanics parameters for both the single rate and multirate schemes. Second, following the approach in Almani et al. (Comput. Geosci. 21:1157–1172 2017), we establish a priori error estimates for the single rate case of the scheme in homogeneous poroelastic media. To the best of our knowledge, this is the first rigorous and complete mathematical analysis of the undrained split iterative coupling scheme in heterogeneous poroelastic media.

Performance Evaluation of Numerical Finite Element Coupled Algorithms for Structure–Electric Interaction Analysis of MEMS Piezoelectric Actuator

This work presents multiphysics numerical analysis of piezoelectric actuators realized using the finite element method (FEM) and their performances to analyze the structure-electric interaction in three-dimensional (3D) piezoelectric continua. Here, we choose the piezoelectric bimorph actuator without the metal shim and with the metal shim as low-frequency problems and a surface acoustic wave device as a high-frequency problem. More attention is given to low-frequency problems because in our application micro air vehicle’s wings are actuated by piezoelectric bimorph actuators at low frequency. We employed the Newmark’s time integration and the central difference time integration to study the dynamic response of piezoelectric actuators. Monolithic coupling, noniterative partitioned coupling and partitioned iterative coupling schemes are presented. In partitioned iterative coupling schemes, the block Jacobi and the block Gauss–Seidel methods are employed. Resonance characteristics are very important in micro-electro-mechanical system (MEMS) applications. Therefore, using our proposed coupled algorithms, the resonance characteristics of bimorph actuator is analyzed. Comparison of the accuracy and computational efficiency of the proposed numerical finite element coupled algorithms have been carried out for 3D structure–electric interaction problems of a piezoelectric actuator. The numerical results obtained by the proposed algorithms are in good agreement with the theoretical solutions.

Adaptive asynchronous time-stepping, stopping criteria, and a posteriori error estimates for fixed-stress iterative schemes for coupled poromechanics problems

In this paper we develop adaptive iterative coupling schemes for the Biot system modeling coupled poromechanics problems. We particularly consider the space–time formulation of the fixed-stress iterative scheme, in which we first solve the problem of flow over the whole space–time interval, then exploiting the space–time information for solving the mechanics. Two common discretizations of this algorithm are then introduced based on two coupled mixed finite element methods in-space and the backward Euler scheme in-time. Therefrom, adaptive fixed-stress algorithms are build on conforming reconstructions of the pressure and displacement together with equilibrated flux and stresses reconstructions. These ingredients are used to derive a posteriori error estimates for the fixed-stress algorithms, distinguishing the different error components, namely the spatial discretization, the temporal discretization, and the fixed-stress iteration components. Precisely, at the iteration k≥1 of the adaptive algorithm, we prove that our estimate gives a guaranteed and fully computable upper bound on the energy-type error measuring the difference between the exact and approximate pressure and displacement. These error components are efficiently used to design adaptive asynchronous time-stepping and adaptive stopping criteria for the fixed-stress algorithms. Numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive iterative coupling algorithms.