Monolithic Coupling Research Articles

A robust 3D finite element framework for monolithically coupled thermo-hydro-mechanical analysis of fracture growth with frictional contact in porous media

This paper presents the formulation of a robust integrated framework for the coupled multiphysics and multiple fracture growth analysis in porous media. The finite element-based thermo-hydro-mechanical method for fracture growth with frictional contact (THMf-g) simultaneously solves monolithically coupled equations, incorporating contact and frictional constraints from fracture sliding. It also implements an adaptive process to update fields and geometries during fracture growth, effectively modeling emerging new surfaces. The thermo-hydro-mechanical fields are derived from fundamental principles of mass, momentum, and energy conservation and discretized numerically using the finite element method. Fractures are represented as sub-dimensional surfaces embedded within the volume of the porous medium. The growth of multiple fracture is modeled based on stress intensity factors at fracture tips, with fracture aperture and permeability emerging as dynamic properties of the system. The main novelty of this work lies in extending the implicitly solved monolithic coupling to include frictional and growth modeling for multiple non-planar fractures of emerging geometry in three dimensions. This includes the direct incorporation of cubic terms in the fracture flow equations and convection terms in the heat transfer equations, adopting an incremental method to solve these coupled, nonlinear equations. To ensure energy conservation, the heat equations are resolved using an implicit scheme, establishing a velocity dependency on pressure fields and introducing quadratic terms into the heat equation. Furthermore, the heat transfer equation has been revised to account for the work done on the fluid, enhancing the accuracy of thermal modeling. A contact mechanics leader–follower strategy tracks a conformal mesh split at each fracture, accounting explicitly for permeability changes during deformation and growth, effectively reducing computational complexity and cost. The iterative process for applying these constraints and the solution of the coupled THMf-g with friction and growth is described. The method does not require calibrated material properties or artificial length scale parameters, and relies on laboratory-measured properties with direct physical interpretation. A detailed algorithm is presented, including the updating of apertures and handling of new surfaces during the simulation, accounting for the variability of fracture permeability and its interplay with contact and friction during the non-linear solution. Several validating numerical experiments are benchmarked against analytical solutions, demonstrating the accuracy and reliability of the proposed framework in capturing complex fracture behavior and multiphysics interactions in porous media.

An automatic simulation pipeline for coupled simulations of acoustic damping materials

AbstractFoamed materials are widely used to reduce noise due to their comparably good acoustic damping behavior. However, out of a large variety of these materials a suitable candidate has to be identified for each application. This is a challenging process that is typically guided by experiments and experience. While numerical simulations could support these experiments and reduce the effort to a great extent, no suitable discretization approach has yet been established that can fully capture the complex geometry of the foam. A fully resolved model is desirable in order to yield reliable predictions that can then be used to establish homogenized models. We established a monolithic coupling approach based on the finite cell method (FCM) that realizes a vibroacoustic simulation in this sense. The fluid and the structure domain are discretized by Cartesian grids and the geometry defined based on computed tomography scans is accounted for during the quadrature of the weak form. Our simulation in the time domain makes use of explicit time marching schemes and is therefore limited by a critical time step size. This is known to be arbitrarily low for discretizations with the FCM containing cells with arbitrarily small support. As a remedy against this we use the classical ‐stabilization technique and investigate its potentials and limitations.

Investigation of fluid added mass matrix during hydroelastic slamming of wedges

The investigation of the fluid added mass (FAM) matrix during the water entry of an elastic structure is crucial for understanding the mechanism of hydroelastic slamming, as it characterizes the fluid–structural interaction process. In this paper, we excavate a model for directly estimating the FAM matrix during hydroelastic slamming and employ this model to study the limitations and optimized values of the relaxation factor for a partitioned coupling solver. Our simulation of hydroelastic slamming couples the Wagner theory with the modal superposition method (MSM), and the FAM matrix is computed using a monolithic coupling scheme. We conduct a series of parametric studies to analyze how the modal number, hydroelasticity, deadrise angle, and structural boundary condition influence the eigenvalues of the FAM matrix. Based on these studies, we then propose an analytical model to directly express the eigenvalues of the FAM matrix, demonstrating their linear relationship with the wetted ratio. Furthermore, we apply this analytical model to analyze the relaxation factor in the partitioned coupling scheme, theoretically explaining its strong connection to the FAM effect. The limitations and optimized values of the relaxation factor are theoretically presented, and these results are verified through our numerical tests.

Efficient multi-scale staggered coupling of discrete and boundary element methods for dynamic problems

This paper presents a novel and highly efficient approach for coupling the Discrete Element Method (DEM) and the Boundary Element Method (BEM) for time-domain simulations of dynamic problems, utilising multi-scale staggered time integration. While the DEM captures phenomena with discontinuous behaviours, such as fracturing and granular flow, the BEM excels in accurately modelling seismic wave propagation in infinite domains. By separately solving the governing equations of the DEM and BEM at different time instants, the proposed scheme considerably enhances computational efficiency compared to conventional monolithic coupling schemes. The incorporation of non-conforming interfaces enables larger time steps in the BEM, thereby reducing computational costs and memory usage. Moreover, an innovative coupling of DEM rotations with the BEM displacement field is introduced, leading to more accurate and realistic modelling of complex dynamics. Numerical experiments are conducted to demonstrate the superior accuracy and efficiency of the proposed method, establishing its potential for modelling a wide range of dynamic problems.

Fluid flow migration, rock stress and deformation due to a crustal fault slip in a geothermal system: A poro-elasto-plastic perspective

Geothermal systems are commonly genetically and spatially associated with volcanic complexes, which in turn, are located nearby crustal fault systems. Faults can alter fluid flow in their surroundings, potentially acting as barriers or conduits for fluids, depending on their architecture and slip-rate. However, this fundamental control on fluid migration is still poorly constrained. Most previous modeling efforts on volcanic and hydrothermal processes consider either only fluid flow in their formulations, or only a mechanical approach, and seldom a full, monolithic coupling between both. In this work, we present a poro-elasto-plastic Finite Element Method (FEM) to address the first-order, time-dependent control that a strike-slip crustal fault exerts on a nearby geothermal reservoir. For the model setting, we selected the Planchón-Peteroa geothermal system in the Southern Andes Volcanic Zone (SAVZ), for which the geometry and kinematics of a potentially seismogenic fault and fluid reservoir is constrained from previous geological and geophysical studies. We assess the emergence and diffusion of fluid pressure domains due to fault slip, as well as the development of tensile/dilational and compressive/contractional domains in the fault' surroundings. Mean stress and volumetric strain magnitudes in these domains range between ±1 [MPa] and ±10−4 [-], respectively. Our results show the appearance of negative and positive fluid pressure domains in these dilational and contractional regions, respectively. We also investigate the spatial and temporal evolution of such domains resulting from changes in fault permeability and shear modulus, fluid viscosity, and rock rheology. These variations in fluid pressure alter the trajectory of the reservoir fluids, increasing migration to the eastern half of the fault, reaching a maximum fluid flux of 8 to 70 times the stationary flux. Pressure-driven fluid diffusion over time causes fluid flow to return to the stationary state between weeks to months after fault slip. These results suggest that the mechanism that exerts a first-order control is similar to a suction pump, whose duration heavily depends on fault permeability and fluid viscosity. We also show how a von Mises plasticity criterion locally enhances fluid flow. The transient process analyzed in this work highlights the importance of addressing the solid-fluid coupling in numerical models for volcano-tectonic studies.

The monolithic coupling of straight tube Coriolis mass flowmeter with liquid hydrogen

The Coriolis mass flowmeter (CMF) has the advantages of a simple structure and direct measurement of fluid mass flow rate. This paper numerically simulates the fluid-structure interaction (FSI) of a straight tube CMF based on the monolithic coupling methods of hydrodynamics and structural mechanics with liquid hydrogen (LH2) as the working fluid. The simulated time lag for the CMF with water is firstly compared with the results from the published experiment and the iterative partitioned coupling method to validate the model. The comparison shows that the results of the monolithic coupling are closer to the experimental results. Then the monolithic coupling method is used to analyze the FSI characteristics of the CMF with LH2. And the general flow-induced vibration results with LH2 are presented for the straight tube CMF, including the dry and wet vibration modes. The velocity and the detector positions on the time lag are specially discussed. The results reveal the FSI characteristics of the straight tube CMF with LH2.

Hydrophobic and Hydrophilic Solid-Fluid Interaction

We propose a novel solid-fluid coupling method to capture the subtle hydrophobic and hydrophilic interactions between liquid, solid, and air at their multi-phase junctions. The key component of our approach is a Lagrangian model that tackles the coupling, evolution, and equilibrium of dynamic contact lines evolving on the interface between surface-tension fluid and deformable objects. This contact-line model captures an ensemble of small-scale geometric and physical processes, including dynamic waterfront tracking, local momentum transfer and force balance, and interfacial tension calculation. On top of this contact-line model, we further developed a mesh-based level set method to evolve the three-phase T-junction on a deformable solid surface. Our dynamic contact-line model, in conjunction with its monolithic coupling system, unifies the simulation of various hydrophobic and hydrophilic solid-fluid-interaction phenomena and enables a broad range of challenging small-scale elastocapillary phenomena that were previously difficult or impractical to solve, such as the elastocapillary origami and self-assembly, dynamic contact angles of drops, capillary adhesion, as well as wetting and splashing on vibrating surfaces.

A Material Law Based on Neural Networks and Homogenization for the Accurate Finite Element Simulation of Laminated Ferromagnetic Cores in the Periodic Regime

Electromagnetic fields and eddy currents in thin electrical steel laminations are governed by the laws of magnetodynamics with hysteresis. If the lamination is large with respect to its thickness, field and current distributions are accurately resolved by solving a 1-D finite element magnetodynamic problem with hysteresis across half the lamination thickness. This 1-D model is able to deliver mesoscopic information to be used, after appropriate homogenization, in the macroscopic modeling of an electrical machine or transformer. As each evaluation of such a homogenized model implies a finite element simulation at the mesoscale, a monolithic coupling might be very time-consuming. This article proposes an alternative approach, assuming a periodic excitation of the system, where the parameters of a parametric homogenized material law are determined in each finite element with a neural network. The local material law can then be used as a conventional constitutive relationship in a 2-D or 3-D modeling, with a massive speed-up with respect to the monolithic coupling.

Parameter-robust methods for the Biot–Stokes interfacial coupling without Lagrange multipliers

In this paper we advance the analysis of discretizations for a fluid-structure interaction model of the monolithic coupling between the free flow of a viscous Newtonian fluid and a deformable porous medium separated by an interface. A five-field mixed-primal finite element scheme is proposed solving for Stokes velocity-pressure and Biot displacement-total pressure-fluid pressure. Adequate inf-sup conditions are derived, and one of the distinctive features of the formulation is that its stability is established robustly in all material parameters. We propose robust preconditioners for this perturbed saddle-point problem using appropriately weighted operators in fractional Sobolev and metric spaces at the interface. The performance is corroborated by several test cases, including the application to interfacial flow in the brain.

A reduced unified continuum formulation for vascular fluid–structure interaction

We recently derived the unified continuum and variational multiscale formulation for fluid–structure interaction (FSI) using the Gibbs free energy as the thermodynamic potential. Restricting our attention to vascular FSI, we now reduce this formulation in arbitrary Lagrangian–Eulerian (ALE) coordinates by adopting three common modeling assumptions for the vascular wall. The resulting semi-discrete formulation, referred to as the reduced unified continuum formulation, achieves monolithic coupling of the FSI system in the Eulerian frame through a simple modification of the fluid boundary integral. While ostensibly similar to the semi-discrete formulation of the coupled momentum method introduced by Figueroa et al., its underlying derivation does not rely on an assumption of a fictitious body force in the elastodynamics sub-problem and therefore represents a direct simplification of the ALE method. Furthermore, uniform temporal discretization of the entire FSI system is performed via the generalized-α scheme. In contrast to the predominant approach yielding only first-order accuracy for pressure, we collocate both pressure and velocity at the intermediate time step to achieve uniform second-order temporal accuracy. In conjunction with quadratic tetrahedral elements, our methodology offers higher-order temporal and spatial accuracy for quantities of clinical interest, including pressure and wall shear stress. Furthermore, without loss of consistency, a segregated predictor multi-corrector algorithm is developed to preserve the same block structure as for the incompressible Navier–Stokes equations in the implicit solver’s associated linear system. Block preconditioning of a monolithically coupled FSI system is therefore made possible for the first time. Compared to alternative preconditioners, our three-level nested block preconditioner, which achieves improved representation of the Schur complement, demonstrates robust performance over a wide range of physical parameters. We present verification of our methodology against Womersley’s deformable wall theory and additionally develop practical modeling techniques for clinical applications, including tissue prestressing. We conclude with an assessment of our combined FSI technology in two patient-specific cases.

Fully 3D ship hydroelasticity: Monolithic versus partitioned strategies for tight coupling

This paper analyzes the partitioned and monolithic strategies to simulate tightly coupled hidroelastic problems. The seakeeping hydrodynamics solver used is based on a first-order linear time-domain FEM model with forward speed and double-body linearization. The structural dynamics solver is based on a full 3D time-domain FEM with corotational shell elements accounting for the geometric non-linearity. Both solvers are implemented under the same programming framework, which allows to implement the monolithic strategy, and to minimize the communication overheads of the partitioned strategy. Two case studies are used to test and compare the partitioned and monolithic coupling: a flexible catamaran in oblique waves, and a large floating reticulated structure made of fiber reinforced plastic. In both cases, the monolithic strategy is between three and four times faster than the partitioned strategy.This project has been developed under the H2020 project FIBRESHIP aimed at developing the technology to design and build the structure of large-length vessels in fiber reinforced polymers.

Chemo-mechanical modelling of swelling and crosslinking reaction kinetics in alginate hydrogels: A novel theory and its numerical implementation

Hydrogels are mechanically stabilized through the action of external agents which induce the formation of crosslinks in the polymer network as a consequence of transport and reactive mechanisms. Crosslinking increases the stiffness of the construct, produces inelastic deformations in the polymer network and interacts with the swelling capacity of hydrogels. The control of this process is hence crucial for fulfilling functional criteria in several technological fields, like drug-delivery or bioprinting. Nevertheless, experimental approaches for monitoring the crosslinking kinetics with the required resolution are currently missing. The development of new computational models in the field might open the way to novel investigation tools. This paper presents a thermodynamically consistent chemo-mechanical model in large deformation for reactive–diffusive mechanisms occurring during crosslinking in alginate hydrogels. The system accounts for shrinking and swelling effects, fluid movements, as well as the reaction kinetics of calcium-induced crosslinking. Crosslinks alter mechanical and diffusive properties in the hydrogel. Moreover, on the basis of thermodynamic arguments, internal stresses directly affect the crosslinking kinetics, revealing a two-way coupling between mechanics and chemistry. The model is implemented in a finite element framework, considering a monolithic coupling between chemical transport and mechanics. The computational framework allows characterizing the (experimentally inaccessible) heterogeneous distribution of mechano-chemical quantities and properties in the hydrogel. Parametric campaigns of simulations are presented to investigate hydrogels’ behaviour and compare numerical outcomes with available experimental evidence.

A Dynamic and Fully Implicit Non‐Isothermal, Two‐Phase, Two‐Component Pore‐Network Model Coupled to Single‐Phase Free Flow for the Pore‐Scale Description of Evaporation Processes

AbstractWe couple a dynamic, fully implicit non‐isothermal two‐phase, two‐component pore‐network model (PNM) to a free‐flow domain where the single‐phase flow Navier‐Stokes equations with component and energy transport are solved. A monolithic coupling scheme guarantees the conservation of mass, momentum and energy across the coupling interface and allows solving the system without any coupling iterations. We present a numerical example and show that the model is able to temporally resolve the pore‐local flow and transport dynamics of an evaporation process.

Computational framework for monolithic coupling for thin fluid flow in contact interfaces

We developed a computational framework for simulating thin fluid flow in narrow interfaces between contacting solids, which is relevant for a range of engineering, biological and geophysical applications. The treatment of this problem requires coupling between fluid and solid mechanics equations, further complicated by contact constraints and potentially complex geometrical features of contacting surfaces. We developed a monolithic finite-element framework for handling mechanical contact, thin incompressible viscous flow and fluid-induced tractions on the surface of the solid, suitable for both one- and two-way coupling approaches. Additionally, we consider the possibility of fluid entrapment in ”pools” delimited by contact patches and its pressurization following a non-linear compressibility constitutive law. Image analysis algorithms were adapted to identify the local status of each interface element (i.e. distinguish between contact, fluid flow and trapped fluid zones) within the Newton–Raphson loop. First, an application of the proposed framework for a problem with a model geometry is given, and the robustness is demonstrated by the residual-wise and status-wise convergence. The full capability of the developed two-way coupling framework is demonstrated on a problem of a fluid flow in contact interface between a solid with representative rough surface and a rigid flat. The evolution of the contact pressure, fluid flow pattern and the morphology of trapped fluid zones under increasing external load until the complete sealing of the interface is displayed. Additionally, we demonstrated an almost mesh-independent result of a refined post-processing approach to the real contact-area computation. The developed framework permits not only to study the evolution of effective properties of contact interfaces, such as transmissivity and real contact area, but also to highlight the difference between one- and two-way coupling approaches, and, in particular, to quantify the effect of trapped fluid ”pools” on the coupled problem.

Spatio-temporal aeroacoustic–structural responses of cavity-backed elastic panel liner exposed to grazing duct flow

An in-duct device laid with acoustic liner is a popular noise control option for extensive ductworks carrying unsteady flow in various engineering applications. Besides the conventional absorptive liner, elastic panel liner backed by a cavity emerges as a promising alternative concept that has advantages in providing effective broadband reduction at low frequencies yet with significantly low fluid dynamic pressure loss. The present paper reports a numerical study of aeroacoustic–structural interaction of elastic panel liner exposed to subsonic duct boundary layer flow in time domain and explores its role in liner mitigation of plane wave broadband acoustic excitation. The aeroacoustics of the duct flow and panel dynamics are respectively modeled by the two-dimensional compressible Navier–Stokes equations that solved by the conservation element and solution element (CE/SE) method, and one-dimensional elastic panel equation that is solved with finite difference method. The interaction between aeroacoustics and panel dynamics is resolved with a monolithic coupling scheme that is well validated down to acoustic scales. Special attention is placed on the sensitivity of liner acoustic performance to duct flow Mach number (M≤0.3) and the relative directions between duct flow and acoustic excitation. Extensive cross-spectral analyses of acoustic results reveal that the transmission loss of the elastic panel liner is primarily a result of the destructive interference of the incident wave with the scattered wave as well as the dissipation of non-planar waves radiated by the panel vibration. The contribution of interference and dissipation to liner reflection and absorption are particularly pronounced at low and high frequencies respectively. In addition, the effect of flow is not strong for M≤0.1. The presence of flow with the incident wave in the flow direction mainly weakens the reflection. However, the effect of flow with incident waves against the flow largely enhances the absorption. Overall, the elastic panel liner is much more effective when the noise source radiates against the flow.

Parameter-Robust Methods for the Biot-Stokes Interfacial Coupling Without Lagrange Multipliers

In this paper we advance the analysis of discretizations for a fluid-structure interaction model of the monolithic coupling between the free flow of a viscous Newtonian fluid and a deformable porous medium separated by an interface. A five-field mixed-primal finite element scheme is proposed solving for Stokes velocity-pressure and Biot displacement-total pressure-fluid pressure. Adequate inf-sup conditions are derived, and one of the distinctive features of the formulation is that its stability is established robustly in all material parameters. We propose robust preconditioners for this perturbed saddle-point problem using appropriately weighted operators in fractional Sobolev and metric spaces at the interface. The performance is corroborated by several test cases, including the application to interfacial flow in the brain.

Effect of Computational Schemes on Coupled Flow and Geo-Mechanical Modeling of CO2 Leakage through a Compromised Well

Carbon capture, utilization, and storage (CCUS) describes a set of technically viable processes to separate carbon dioxide (CO2) from industrial byproduct streams and inject it into deep geologic formations for long-term storage. Legacy wells located within the spatial domain of new injection and production activities represent potential pathways for fluids (i.e., CO2 and aqueous phase) to leak through compromised components (e.g., through fractures or micro-annulus pathways). The finite element (FE) method is a well-established numerical approach to simulate the coupling between multi-phase fluid flow and solid phase deformation interactions that occur in a compromised well system. We assumed the spatial domain consists of a three-phases system: a solid, liquid, and gas phase. For flow in the two fluids phases, we considered two sets of primary variables: the first considering capillary pressure and gas pressure (PP) scheme, and the second considering liquid pressure and gas saturation (PS) scheme. Fluid phases were coupled with the solid phase using the full coupling (i.e., monolithic coupling) and iterative coupling (i.e., sequential coupling) approaches. The challenge of achieving numerical stability in the coupled formulation in heterogeneous media was addressed using the mass lumping and the upwinding techniques. Numerical results were compared with three benchmark problems to assess the performance of coupled FE solutions: 1D Terzaghi’s consolidation, Liakopoulos experiments, and the Kueper and Frind experiments. We found good agreement between our results and the three benchmark problems. For the Kueper and Frind test, the PP scheme successfully captured the observed experimental response of the non-aqueous phase infiltration, in contrast to the PS scheme. These exercises demonstrate the importance of fluid phase primary variable selection for heterogeneous porous media. We then applied the developed model to the hypothetical case of leakage along a compromised well representing a heterogeneous media. Considering the mass lumping and the upwinding techniques, both the monotonic and the sequential coupling provided identical results, but mass lumping was needed to avoid numerical instabilities in the sequential coupling. Additionally, in the monolithic coupling, the magnitude of primary variables in the coupled solution without mass lumping and the upwinding is higher, which is essential for the risk-based analyses.

Finite element formulation to study thermal stresses in nanoencapsulated phase change materials for energy storage

Nanoencapsulated phase change materials (nePCMs) – which are composed of a core with a phase change material and of a shell that envelopes the core – are currently under research for heat storage applications. Mechanically, one problem encountered in the synthesis of nePCMs is the failure of the shell due to thermal stresses during heating/cooling cycles. Thus, a compromise between shell and core volumes must be found to guarantee both mechanical reliability and heat storage capacity. At present, this compromise is commonly achieved by trial and error experiments or by using simple analytical solutions. On this ground, the current work presents a thermodynamically consistent and three-dimensional finite element (FE) formulation considering both solid and liquid phases to study thermal stresses in nePCMs. Despite the fact that there are several phase change FE formulations in the literature, the main novelty of the present work is its monolithic coupling – no staggered approaches are required – between thermal and mechanical fields. Then, the FE formulation is implemented in a computational code and it is validated against one-dimensional analytical solutions. Finally, the FE model is used to perform a thermal stress analysis for different nePCM geometries and materials to predict their mechanical failure by using Rankine’s criterion.

Interfacial coupling across incompatible meshes in a monolithic finite-strain thermomechanical formulation

A stabilized interface method is presented for monolithic coupling of thermomechanical fields across incompatible internal interfaces. Embedding Discontinuous Galerkin (DG) ideas in the Variational Multiscale (VMS) framework yields interface coupling terms that accommodate nonmatching discretizations across glued meshes. A unique feature of the method that emanates from fine-scale modeling facilitated by VMS is the derivation of analytical expressions for the Lagrange multipliers that enforce continuity of the fields across nonmatching meshes. This stabilized form is free of any user-defined parameters and renders itself to consistent linearization that leads to quadratic convergence of the method. The coupling terms at the interface also provide an avenue to employ various kinematic models for interfacial response in multi-constituent materials. An interfacial gap/crack function is introduced together with the conjugate traction vector to model discrete interfacial failure in multi-material applications. An algorithm for consistent implementation of interfacial models for tension, compression, and sliding friction across non-conforming meshes is presented. Test cases that employ 8-node hexahedra and 4-node tetrahedra are presented that highlight the generality of the method and validate its robustness for mathematically nonsmooth thermomechanical problems.

A least‐squares finite element approach to model fluid‐structure interaction problems

AbstractIn this contribution an approach to model fluid‐structure interaction (FSI) problems with monolithic coupling is presented. The fluid as well as the structural domain are discretized using the least‐squares finite element method (LSFEM), whose application results in a minimization problem with symmetric positive definite systems also for non self‐adjoint operators, see e.g. [2]. In this study, the second‐order systems are reduced to first‐order systems by introducing new variables, which leads to least‐squares formulations for both domains based on the stresses and velocities as presented in e.g. [5] and [7]. A conforming discretization of the unknown fields in H1 and H(div) using Lagrange interpolation polynomials and vector‐valued Raviart‐Thomas interpolations functions, respectively, leads to the inherent fulfillment of the FSI coupling conditions. In more detail, a discretization in H1 ensures continuity of the velocity field and a discretization in H(div) results in continuity of the normal stress components at the interface.