Implicit Time-marching Scheme Research Articles

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.

Contact dynamic analysis of tether-net system for space debris capture using incremental potential formulation

An efficient computational method is proposed to combine the discrete differential geometry (DDG) method with the multi-dimensional incremental potential (IP) formulation to simulate the contact dynamics between a shooting net and a defunct satellite. The geometrically nonlinearity of the elastic net is described using the DDG method, which discretizes the flexible net into multiple nodes and connected edges. The internal elastic forces and the associated Jacobians are derived analytically. The multi-dimensional incremental potential formulation is used to achieve the intersection-free condition during the physical collision between deformable tether-net and rigid space debris. The smooth IP barrier functional constructed for contact simulation is differentiable, such that the nonlinear elastodynamic system can be robustly solved through a fully implicit time-marching scheme. Our numerical approach is first applied to simple examples, and demonstrating its effectiveness in dealing with non-penetration contacts of elements with different dimensions, such as ‘edge-to-edge’ and ‘edge-to-surface’. Next, the dynamic interaction between the deploying net and defunct satellite is analysed. A parameter sweep is performed with our robust and efficient simulator to show the configuration evolution of the net shot from different relative locations of the target object. Our numerical results can provide an optimal deployment strategy for tether-net capture systems during the active debris removal missions.

Urban Heat Island Dynamics in an Urban–Rural Domain with Variable Porosity: Numerical Methodology and Simulation

Heat transfer and fluid dynamics modeling in porous media is a widely explored topic in physics and applied mathematics, and it involves advanced numerical methods to address its non-linear nature. One interesting application has been the urban-heat-island (UHI) numerical simulation. The UHI is a negative consequence of the increasing urbanization in cities, which is defined as the presence of warm temperatures inside the urban canopy in contrast to the colder surroundings. Furthermore, an interesting phenomena occurs within a UHI context when the city transitions from a heat island to a cold island, matching the increases and decreases of solar radiation over the span of a day, as well as the decrease in the UHI intensity as a result of wind action. The numerical study in this paper had, as its main goal, to reproduce this phenomenon. Therefore, the key elements proposed in this work were the following: A 2D horizontal urban–rural domain that had a variable porosity with a Gaussian distribution centered in the city center. A non-stationary Darcy–Forchheimer–Brinkman model to simulate the flow in porous media, combined with an air–soil heat transport model linked by a balancing equation for the surface energy that includes the evapotranspiration of plants. In regards to the numerical resolution of the model, a classical numerical methodology based on the finite elements of Lagrange P1 type combined with explicit and implicit time-marching schemes have been effective for high-quality numerical simulations. Several numerical tests were performed on a domain inspired by the metropolitan region of Guadalajara (Mexico), in which not only the temperature inversion was reproduced but also the simulation of the UHI transition by strong wind gusts.

Fast and Accurate Core Analysis by the Full-Immersion Pressure-Pulse Decay: Part 1—Theory

Summary Core-scale measurements are considered the ground truth that oil and gas or electric utility companies use to predict the migration of fluids such as oil, natural gas, carbon dioxide, or brine deep underground during their extraction or injection operations. To provide a greater understanding of petrophysical properties of low-permeability geologic formations such as shales and tight gas sandstones, this study introduces a novel core-analysis procedure. The technique follows conventional pressure-pulse-decay permeametry, where the pressure in an inlet chamber adjacent to a cylindrical core plug undergoes a rapid pressurization, the system is shut in, and the pressure reaches a new equilibrium. However, unlike a standard unidirectional pressure-pulse decay, the full-immersion pressure-pulse decay (Hannon 2019) applies a pressure disturbance to the entire outer surface area of the sample. This article covers the numerical simulator designed to model flow through an anisotropic porous sample in this scenario. The model assumes distinct but uniform permeabilities along the radial and axial directions of the cylindrical plug sample. When extracting a plug vertically (or perpendicular to bedding), the permeability along the radial direction associates with the horizontal permeability (i.e., parallel to bedding), whereas in the axial direction, flow occurs perpendicular to bedding (a vertical permeability). Investigations of these model outputs demonstrate an approximately 20-fold decrease in time to complete a full-immersion experiment compared with conventional pressure-pulse decay. Furthermore, the pressure-decay curves resulting from the full-immersion method have slightly different shapes than those resulting from other unidimensional transient methods. These differences begin to demonstrate that, under achievable experimental conditions, the analysis of pressure data from one full-immersion test could enable the simultaneous estimation of the apparent permeabilities parallel and perpendicular to bedding of a cylindrical sample in addition to its porosity. A follow-up article finalizes the proof of this capability with a parameter-estimation procedure and presents experimental verification through a proof-of-concept study. Described in greater detail in that article, the parameter estimator requires multiple forward simulations to analyze the data, which behooves minimizing the compute time for each simulation. By using an alternating direction implicit time-marching scheme and a structured but variably spaced grid, the numerical simulator built for this purpose provides a forward model output with suitable accuracy in approximately 0.5 seconds.

Advantages of static condensation in implicit compressible Navier–Stokes DGSEM solvers

We consider implicit time-marching schemes for the compressible Navier-Stokes equations, discretised using the Discontinuous Galerkin Spectral Element Method with Gauss-Lobatto nodal points (GL-DGSEM). We compare classic implicit strategies for the full Jacobian system to our recently developed static condensation technique for GL-DGSEM Rueda-Ramírez et al.(2019), A Statically Condensed Discontinuous Galerkin Spectral Element Method on Gauss-Lobatto Nodes for the Compressible Navier-Stokes Equations [1]. The Navier-Stokes system is linearised using a Newton-Raphson method and solved using an iterative preconditioned-GMRES solver. Both the full and statically condensed systems benefit from a Block-Jacobi preconditioner.We include theoretical estimates for the various costs involved (i.e. calculation of full and condensed Jacobians, factorising and inverting the preconditioners, GMRES steps and overall costs) to clarify the advantages of using static condensation in GL-DGSEM, for varying polynomial orders. These estimates are then examined for a steady three-dimensional manufactured solution problem and for an two-dimensional unsteady laminar flow over a NACA0012 airfoil. In all cases, we test the schemes for high polynomial orders, which range from 2 to 8 for a manufactured solution case and from 2 to 5 for the NACA0012 airfoil. The statically condensed system shows computational savings, which relate to the smaller system size and cheaper Block-Jacobi preconditioner with smaller blocks and better polynomial scaling, when compared to the preconditioned full Jacobian system (not condensed). The advantage of using static condensation is more noticeable for higher polynomial orders.

Staggered-grid entropy-stable multidimensional summation-by-parts discretizations on curvilinear coordinates

The entropy conservative/stable staggered grid tensor-product algorithm of Parsani et al. [1] is extended to multidimensional SBP discretizations. The required SBP preserving interpolation operators are proven to exist under mild restrictions and the resulting algorithm is proven to be entropy conservative/stable as well as elementwise conservative. For 2-dimensional simplex elements, the staggered grid algorithm is shown to be more accurate and have a larger maximum time step restriction as compared to the collocated algorithm. The staggered algorithm significantly reduces the number of (computationally expensive) two-point flux evaluations, which is potentially important for both explicit and implicit time-marching schemes. Furthermore, the staggered algorithm requires fewer degrees of freedom for comparable accuracy, which has favorable implications for implicit time-marching schemes.

Computer simulation of hyperthermia with nanoparticles using an OcTree finite volume technique

This paper develops an efficient finite volume approach using OcTree mesh for simulation of hyperthermia treatment by injection of nanoparticles. A three-dimensional nonlinear Pennes' model is employed to simulate the heat transfer in a heterogeneous blood perfused living tissue. The OcTree mesh is a good compromise between regular grid from finite difference method and unstructured meshes from finite element method due to the possibility of applying local refinement using a nonconforming mesh in a straightforward manner. Implicit time-marching scheme coupled with the Picard method and explicit method are used. A numerical example is analyzed, illustrating the advantages of the approach.

Partially Implicit Method for Fast Magnetization Analysis Using Assembled Domain Structure Model

A partially implicit solution method is developed for fast magnetization analysis using an assembled-domain-structure model. The implicit time-marching scheme is simplified to avoid the inversion of a dense Jacobian matrix by separating the magnetostatic field into near and far components. The proposed approach achieves a reduction of computation time by 94%-99%.

Acoustic wave generation in near-critical supercritical fluids: Effects on mass transfer and extraction

The supercritical fluid extraction (SFE) process has attracted increasing interest over the recent decades. This is particularly motivated by concerns regarding the environmental effects of using traditional organic (potentially toxic) solvents. However, the conventional process of extraction using supercritical solvents has a slow dynamics even when solute free solvent is re-circulated and therefore improvements in the extraction process are required. The use of acoustic waves represents a potential efficient way of enhancing mass transfer processes in the supercritical region. Mass transfer enhancement using high amplitude resonant acoustic waves is investigated in this study. To carry out the investigation, three different cases are simulated. Acoustically augmented flow and transport in supercritical fluid (CO2) generated by resonant standing wave in a cylindrical shaped resonator (case 1) is simulated first. Based on the knowledge gathered from the simulations of the first case, simulations of acoustically augmented flow and transport in supercritical extraction systems, namely – in a fixed bed extractor (case 2) and membrane contactor (case 3) are also carried out. The oscillatory flow fields in these cases are created by the vibration of one of the end walls of the systems. A real-fluid model for representing the thermo-physical and transport properties of the supercritical fluid is considered. The fully compressible form of the Navier–Stokes equations is used to model the flow fields and an implicit time-marching scheme is used to solve the equations. Due to diverging thermo-physical properties of supercritical fluid near the critical point, large scale oscillations are generated even for small sound field intensity. The effects of near-critical property variations on the formation process of the streaming structures are also investigated. Predicted results from these simulations confirm that acoustic waves significantly accelerate the kinetics of the supercritical extraction process, especially in the near-critical region of operation and improve the final extraction yield. These improvements are attributed to an increase in the overall convective mass transfer coefficient.

A Parallel Adaptive Newton-Krylov-Schwarz Method for 3D Compressible Inviscid Flow Simulations

A parallel adaptive pseudo transient Newton-Krylov-Schwarz (αΨNKS) method for the solution of compressible flows is presented. Multidimensional upwind residual distribution schemes are used for space discretisation, while an implicit time-marching scheme is employed for the discretisation of the (pseudo)time derivative. The linear system arising from the Newton method applied to the resulting nonlinear system is solved by the means of Krylov iterations with Schwarz-type preconditioners. A scalable and efficient data structure for theαΨNKS procedure is presented. The main computational kernels are considered, and an extensive analysis is reported to compare the Krylov accelerators, the preconditioning techniques. Results, obtained on a distributed memory computer, are presented for 2D and 3D problems of aeronautical interest on unstructured grids.

A fully implicit wetting–drying method for DG-FEM shallow water models, with an application to the Scheldt Estuary

Resolving the shoreline undulation due to tidal excursion is a crucial part of modelling water flow in estuaries and coastal areas. Nevertheless, maintaining positive water column depth and numerical stability has proved out to be a very difficult task that requires special attention. In this paper we propose a novel wetting–drying method in which the position of the sea bed is allowed to fluctuate in drying areas. The method is implemented in a Discontinuous Galerkin Finite Element Model (DG-FEM). Unlike most methods in the literature our method is compatible with fully implicit time-marching schemes, thus reducing the overall computational cost significantly. Moreover, global and local mass conservation is guaranteed which is crucial for long-term environmental applications. In addition consistency with tracer equation is also ensured. The performance of the proposed method is demonstrated with a set of test cases as well as a real-world application to the Scheldt Estuary. Due to the implicit time integration, the computational cost in the Scheldt application is reduced by two orders of magnitude. Although a DG-FEM implementation is presented here, the wetting–drying method is applicable to a wide variety of shallow water models.

Eulerian description of high-order bounce-back scheme for lattice Boltzmann equation with curved boundary

We propose an Eulerian description of the bounce-back boundary condition based on the high-order implicit time-marching schemes to improve the accuracy of lattice Boltzmann simulation in the vicinity of curved boundary. The Eulerian description requires only one grid spacing between fluid nodes when second-order accuracy in time and space is desired, although high-order accurate boundary conditions can be constructed on more grid-point support. The Eulerian description also provides an analytical framework for several different interpolation-based boundary conditions. For instance, the semi-Lagrangian, linear interpolation boundary condition is found to be a first-order upwind discretization that changes the time-marching schemes from implicit to explicit as the distance between the fluid boundary node and the solid boundary increases.

A distributed hydro-environmental watershed model with three-zoned cell profiling

A cell-based distributed watershed model is developed which enables us to simulate the hydrological and hydraulic aspects of the watershed in a refined fashion. With three-zoned cell profiling, the model is composed of three sub-models; tank model for a surface water zone, soil moisture model for a surface soil zone, and unconfined shallow groundwater flow model for a subsurface zone. Inclusion of the soil moisture sub-model modified to reroute the infiltration, routed from the tank sub-model, into the return flow and the groundwater recharge features the model. The groundwater flow sub-model, numerically approximated by use of the finite volume method and the implicit time-marching scheme, considers a network of on-farm drainage canals as internal boundaries, which is an essential need for modeling the watershed including farmlands. Cascade-linking of the three sub-models in a cell and assembling of all the cells over the entire watershed domain provides the global equations system to be solved. Applicability of the model is demonstrated with its practical application to a real watershed in that paddy and upland crop fields take great part of the land-use practice. It is then indicated in a quantified manner that rice farming significantly contribute as a major groundwater recharger in an irrigation period to fostering and conservation of regional water resources. Along with appropriately profiling a cell, the model is so versatile and tough that it can be applied without difficulty to a watershed of diverse terrains and land-uses and the computations can stably be carried out. It is thus concluded that the model presently developed could be a powerful “watershed simulator” to investigate and assess the time-varying hydro-environmental properties of a watershed while separating and integrating the hydrological and hydraulic components of particular importance.

A Single-Boundary Implicit and FFT-Accelerated Time-Domain Finite Element-Boundary Integral Solver

A novel time-domain finite-element boundary integral (FE-BI) solver for analyzing broadband scattering and radiation from free-standing electromagnetically large and perfect electrically conducting platforms supporting inhomogeneous and geometrically intricate structures is presented. The solver has three distinctive features that render it especially attractive for broadband analysis of installed antennas. i) The FE and BI solver components are hybridized using a single-surface interface. ii) The hybrid equations are solved by an implicit time-marching scheme accelerated by an (outer) Jacobi iterative solver that leverages (inner) direct FE and iterative BI solvers. iii) The BI solver component is accelerated by a distributed memory parallel implementation of the time-domain adaptive integral method based on the message-passing interface. The accuracy, late-time stability, and performance of the proposed time-domain FE-BI solver are demonstrated via its application to various scattering and radiation problems; moreover, the solver is used to characterize conformally mounted antennas on several platforms including an aircraft

Comparative efficiencies of three parallel algorithms for nonlinear implicit transient dynamic analysis

The work reported in this paper is motivated by the need to develop portable parallel processing algorithms and codes which can run on a variety of hardware platforms without any modifications. The prime aim of the research work reported here is to test the portability of the parallel algorithms and also to study and understand the comparative efficiencies of three parallel algorithms developed for implicit time integration technique. The standard message passing interface (MPI) is used to develop parallel algorithms for computing nonlinear dynamic response of large structures employing implicit time-marching scheme. The parallel algorithms presented in this paper are developed under the broad framework of non-overlapped domain decomposition technique. Numerical studies indicate that the parallel algorithm devised employing the conventional form of Newmark time integration algorithm is faster than the predictor-corrector form. It is also accurate and highly adaptive to fine grain computations. The group implicit algorithm is found to be extremely superior in performance when compared to the other two parallel algorithms. This algorithm is better suited for large size problems on coarse grain environment as the resulting submeshes will obviously be large and thus permit larger time steps without losing accuracy.

A study of the numerical dispersion relation for the 2-D ADI-FDTD method

This letter presents a numerical dispersion relation for the two-dimensional (2-D) finite-difference time-domain method based on the alternating-direction implicit time-marching scheme (2-D ADI-FDTD). The proposed analytical relation for 2-D ADI-FDTD is compared with those relations in the previous works. Through numerical tests, the dispersion equation of this work was shown as correct one for 2-D ADI-FDTD.

Finite volume analysis of dynamic fracture phenomena. I. A node release methodology

A novel node release numerical methodology based on the Finite Volume discretisation (Ivankovic et al., 1994) has been developed and applied for the simulation of (non-linear) elastic fractures where crack propagation velocities are not a priori known. The analysis is performed in two-dimensions by means of an unconditionally stable implicit time-marching scheme and a second order accurate conservative spatial differencing scheme. Application to single crack propagation problems has demonstrated the validity and accuracy of the proposed formulation for dynamic crack propagation. The results obtained by two examples of crack propagation in single edge notch tensile (SENT) and double cantilever beam (DCB) specimens show good agreement with selected numerical, experimental and analytical results. Based on these results, the formulation appears to be well suited to the study of rapid crack propagation (RCP) phenomena in brittle materials. This, combined with the simplicity and accuracy of the formulation, may provide a solid foundation for more physically realistic RCP computational methodologies other than that of the single linear elastic crack representation considered here.

Chebyshev series solution to non-linear boundary value problems in rectangular domain

The von Kármán equations governing the behavior of moderately large deformations of rectangular plates are expressed in displacement field. A methodology based on Chebyshev polynomials approximation to analyze the non-linear boundary value problems in rectangular domain is developed. These non-linear partial differential equations of motion are linearized using quadratic extrapolation techniques. The inertia and dissipative terms are evaluated by employing Houbolt implicit time-marching scheme. The spatial discretization of the differential equations generates incompatibility, viz. greater number of equations than the unknowns. The multiple linear regression analysis, based on the least-square error norm, is employed to overcome the incompatibility and a compatible solution is obtained. Convergence study has been carried out. The clamped and simply supported immovable rectangular plates subjected to static and dynamic loadings are analyzed. Results have been compared with the results obtained by other numerical and analytical methods.

COMPUTATION OF FLOW AND HEAT TRANSFER AROUND A VERTICAL DISCRETE PROTRUDING HEATER USING AN OPERATOR-SPLITTING ALGORITHM

Steady mixed convection around a heated block placed in a vertical channel is numerically studied. The block aspect ratio is taken as I / 3. The following range of parameters has been considered: 200 ≤ Re.≤ 1000 and Gr = 0 and 105. The governing equations have been solved using an operator-splitting technique. This method splits the equations into advection, diffusion, and source terms and solves each component in an implicit time-marching scheme as accurately as possible. The results obtained using this method have been compared with those obtained using the second upwind procedure for the advection terms. The overall distribution of the local flow quantities around the heater is seen to be quite similar for the two methods. However, the average Nusselt number computed using the two methods has significant differences. The operator-splitting method shows that the mixed convection Nusselt number can fall below that of the forced convection value for a certain range of parameters. A comparison of CPU times...

Fourth order finite difference and multigrid methods for modeling instabilities in flat plate boundary layers—2-D and 3-D approaches

An efficient spatial approach which includes a fourth-order finite difference technique on stretched and staggered grids, a fully implicit time-marching scheme, and a semi-coarsening multigrid method based on a modified form of distributive relaxation is developed for solving the transition problem. An improved outflow boundary treatment is also presented that needs only a very short buffer domain to damp all order-one wave reflections. As a first application, we investigate the instability of 2-D incompressible flows over a semi-infinite flat plate by tracking the growth of the stable mode for Tollmien-Schlichting waves. Numerical results show good agreement with the analytic solution obtained from linear stability theory (LST) for the parallel base with small disturbance case. For the non-parallel base flow case, the results are in good agreement with other direct numerical simulations (DNS). An extended 3-D code is then tested as the second applications. Again, good agreement is observed between our approach and LST in the case of small disturbance. Finally, the secondary instability is successfully simulated with moderate grid size and reasonable CPU costs.