Formulation Of Boundary Conditions Research Articles

MultiRegionFoam: A Unified Multiphysics Framework for Multi-Region Coupled Continuum-Physical Problems

Abstract This paper presents a unified framework, called multiRegionFoam, for solving multiphysics problems of the multi-region coupling type within OpenFOAM (FOAM-extend). It is intended to supersede the existing solver with the same name. The design of the new framework is modular, allowing users to assemble a multiphysics problem region-by-region and coupling conditions interface-by-interface. The present approach allows users to choose between deploying either monolithic or partitioned interface coupling for each individual transport equation. The formulation of boundary conditions is generalised in the sense that their implementation is based on the mathematical jump/transmission conditions in the most general form for tensors of any rank. The present contribution focuses on the underlying mathematical model for interface coupled multiphysics problems, as well as on the software design and resulting code structure that enable a flexible and modular approach. Finally, deployment for different multi-region coupling cases is demonstrated, including conjugate heat, multiphase flows and fuel-cells. Source code: multiRegionFoam v1.1 [1], repository https://bitbucket.org/hmarschall/multiregionfoam/. Article highlights A novel multiphysics framework, called multiRegionFoam, has been developed for solving multi-region coupled problems in OpenFOAM. The design of the framework allows for a modular multiphysics setup with freedom of choice on the coupling strategy (partitioned vs. monolithic). Extension of the general transport equation by interface conditions enables a unified coupling approach.

Boundary conditions in flutter simulations of subsonic, transonic and supersonic blade cascades

Simulations of blade flutter are highly sensitive to undesired wave reflections at inlet and outlet boundaries. A careful treatment of boundary conditions is required to prevent the generation of perturbations. This study is motivated by the need to perform flutter analysis of low-pressure steam turbine blades, for which supersonic inflow conditions may occur in the near-tip region. The exact steady non-reflecting boundary condition (NRBC), the spectral NRBC and a simple isentropic boundary condition are implemented in a time-marching flow solver and applied to turbomachinery flutter simulations covering a wide range of operating conditions. For the first time, the spectral NRBC is applied to a blade flutter simulation with a supersonic inlet and its performance is analysed and compared with other boundary condition formulations. It is shown that an effective non-reflective treatment in the design of the boundary condition is essential for an accurate aeroelastic prediction at all operating conditions, including the subsonic flow regime. The limitation of the exact steady NRBC to spatial modes causes it to perform poorly in some unsteady flow simulations, whereas the spectral NRBC achieves a satisfactory suppression of undesired wave reflections in all investigated cases.

A rigorous formulation of drain boundary conditions for groundwater flow modeling in geotechnical engineering

A rigorous formulation of drain boundary conditions for groundwater flow modeling in geotechnical engineering

Microstructure Modeling of Transport and Kinetics in Solid State Batteries with LLZO Solid Electrolyte Fabricated Using the Freeze-Tape-Casting Method

In traditional Li-ion batteries with liquid electrolyte, the transport of lithium ions in the electrolyte and lithium in the active material within a porous electrode are studied extensively. The continuum-scale modeling of these transport phenomena considers a homogenized simulation domain consisting of liquid electrolyte and solid active material with effective transport properties accounting for the heterogeneity [1]. There have been fewer studies that consider digitized versions of the experimentally imaged microstructure in continuum-scale modeling studies, particularly to study mechanics [2]. Despite of the computational challenges, there are inherent advantages in using fully resolved microstructure modeling to improve fundamental understanding of the relevant physical and chemical phenomena in porous electrodes, such as localized degradation and stress concentration. In the case of solid electrolytes, specifically scaffold-type LLZO solid electrolytes, microstructure modeling is of even more relevance, particularly to study anisotropic transport, localized degradation phenomena, stress and deformation leading to loss of contact, and chemo-mechanics. Thus far, microstructure modeling studies on experimentally obtained microstructure consisting of solid electrolyte and active material have been quite limited [3]. The studies that have explored this realm have employed voxel-based mesh to discretize the computation domain [3]. While these methods are sufficient proof of concept, the lack of precision in defining the interface separating active material and electrolyte when using voxel-based mesh provides a source of potentially significant numerical error. Without a highly resolved interface mesh, the total interfacial area increases artificially which leads to inaccuracies in calculations of local reaction rate and overpotential. A highly resolved, smooth interface is also necessary when simulating mechanical deformation and stress field in the microstructure. In the present investigation, we model mass transport and charge transport in the active material/electrolyte microstructure to simulate concentration and potential fields with varying mesh types in order to compare their effectiveness and characterize the error that can be attributed to the mesh type chosen, particularly in the case of voxel-based meshes. Additionally, we study the effect of microstructure domain length chosen in the two directions perpendicular to the shortest length (electrode thickness) as well as boundary condition formulation on the boundaries in these axes on the simulation results. The microstructures studied in this work are derived from the image stacks of the bi/trilayer LLZO solid electrolyte fabricated using freeze-tape-casting method [4]. Overall, the present work aims to develop a better understanding of the best practices in microstructure modeling, particularly when using microstructure image stacks obtained from fabricated samples.

Thermodynamics of Irreversible Processes: Fundamental Constraints, Representations, and Formulation of Boundary Conditions

Starting from the analysis of the lack of positivity of the Cattaneo heat equation, this work addresses the thermodynamic relevance of the positivity constraint in irreversible thermodynamics, that is at least as significant as the entropic constraints. The fulfillment of this condition in hyperbolic models leads to the parametrization of the concentration fields with respect to internal variables associated with the microscopic dynamics. Using Brownian motion theory as a landmark example for deriving macroscopic transport equations from the equations of motion at the particle/molecular level, we discuss two typical problems involving hydrodynamic interactions at the microscale: surface chemical reactions at a solid interface of a diffusing reactant, and mass-balance equations in a complex viscoelastic fluid, in which the physics of the interaction leads either to overcoming the parabolic diffusion model or to considering the parametrization of the concentration with respect to the degrees of freedom associated with the relaxation dynamics of the solvent fluid.

Comparison of Computational Methods and Approaches Applied in Formulation of Boundary Conditions in Lagrange’s Ballistic Problem

This paper compares algorithms for constructing a differential solution to the gas dynamics equations in the surrounding of a moving boundary. In order to compare the algorithms, the Lagrange problem, also known as the piston problem, was chosen as a test problem. A comparison was made between the author’s algorithm based on the method of characteristics and the classical approach with a fictitious cell using the solution of the Riemann problem. Calculations were carried out for the case of subsonic and supersonic gas motion. Comparisons were made on a fixed grid with a dynamically expanding cell (Euler grid) and on a uniformly expanding grid (Arbitrary Lagrangian-Eulerian – ALE grid). The problem was solved using numerical schemes of second order in time and space Lax - Wendroff type: the Richtmyer scheme and the MacCormack scheme. The results of calculations using the characteristics method were used as a benchmark solution. The comparative analysis carried out allows conclusions to be drawn regarding the accuracy of the individual approaches, as well as the scope of their applicability.

Force relaxation response in the poroelastic axisymmetric Boussinesq problem for an indenter of arbitrary profiles I: Permeable and impermeable surface drainage conditions

In this study, we have derived a theoretical solution for the poroelastic axisymmetric Boussinesq problem when an indenter with arbitrary profiles is subjected to step displacement loading. The analysis is conducted within the framework of Biot’s theory, employing the McNamee–Gibson displacement function method. We explore two distinct scenarios for surface drainage boundary conditions: one where the surface allows full permeability and another where it is fully impermeable. The formulation of the mechanical boundary condition at the surface relies on an equation that establishes the relationship between the depth of indentation and the contact radius. Our solution encompasses two aspects: force asymptotes at the undrained and drained limits, and a normalized force relaxation that accounts for transient responses. Specific findings are presented for three indenter shapes: paraboloidal, conical, and cylindrical. Of particular significance, we demonstrate that for these specific indenter shapes, the normalized force relaxation at ω=0 can be expressed in closed form in the Laplace domain. In addition to this analytical breakthrough, we conducted numerical simulations employing actual indenters. Remarkably, the normalized force relaxation curves from actual indenters show negligible dependence on material properties and closely align with our closed-form solution at ω=0. This finding suggests that the closed-form solution could act as a universal master curve capable of characterizing the normalized force relaxation response in instances of a real indenter, irrespective of material properties.

The Einstein equation solution inside a ball with uniform density

A great number of solutions of the Einstein field equation are known. They describe the gravitational field in the empty space-time, in the space-time with electromagnetic field and for a ball filled with a liquid under pressure. The present work is devoted to gravitational field generated by some mass distribution. One of the simplest cases is considered, when mass is uniformly distributed inside a ball and is not moving. The boundary problem for the Einstein equation is formulated. Solution outside the ball is the Schwartzschild solution in vacuum. The coordinates at which the Schwartzschild solution is written are different from the coordinates used in equations for components of the metric tensor inside the ball. Relations between internal and external coordinates are found on the ball surface. They allow to use the Schwartzschild solution for formulation of boundary conditions for internal solution. The solution of the boundary problem is found for the case of weak field. This solution can be used as an example in the analysis of laws of conservation for the gravitational field, in which interaction of mass with field generated by the mass gives a contribution to momentum and energy of the gravitational field.

A highly robust gas network simulation approach through an inherently solvable problem formulation for network states far from intended design points

Pipeline networks are an efficient and widespread transportation system for supplying natural gas as well as increasingly green gas to Europe. However, they are exposed to risks arising from environmental factors, accidents, crime and political issues. This work contributes to the assessment of gas transmission networks and enables a targeted improvement of their resilience. Therefore, the objective of this work is to develop a simulation tool that enables gas transmission system operators (TSOs) to identify weak spots, e.g. potential bottle necks, in their pipeline networks. To this end, a steady-state gas network simulation approach is developed that provides fast results for gas supply deficiencies for complete sets of multi-event disruption scenarios, enabling the identification of the most important network elements. This paper describes how to implement a computationally fast, numerically robust, and physically accurate gas network simulation and assessment tool. By solving mass and momentum balance equations, it accounts for key network elements such as flow control valves, pressure control valves, compressor stations, gas sources, and gas consumers. The central feature that distinguishes this approach from classical steady state solvers is the robustness of the gas flow calculation through the choice of a problem boundary condition formulation, which results in an inherently solvable system of equations. This feature is key in ensuring mathematical convergence of predictions for gas network states that strongly deviate from their original design point, where boundary conditions must not become mathematically invalid but need to reflect physically reasonable behavior. The capabilities of the system are demonstrated using a fictitious gas network and a representative national gas transmission network. Finally, the limits of applicability are outlined based on the size of the network under consideration and the types of analyses.

Study on axial response of semi-parallel wire suspender with spherical plain bearings

Semi-parallel wire suspenders with spherical plain bearings have untwisting under axial force, and the collision in the untwist process bring adverse effects on the structure. To investigate the axial mechanical response of semi-parallel wire suspenders with spherical plain bearings, the equilibrium equations of suspender are deduced by assuming the distribution of steel wire after deformation, the forms of boundary conditions are distinguished according to the use of spherical plain bearing, and the linear equation expressed by equivalent stiffness coefficients is deduced. The Costello theory and finite element modeling are used to verify the proposed calculation method in this paper, and the influence of suspender parameters on the stiffness equivalent coefficient is analyzed. The results show that the proposed calculation method is suitable for multi-layer semi-parallel wire suspenders; the twist angle and the total number of layers of wires have the greatest effect on the stiffness equivalence factor; when the twisting angle is close to 0, the torsional stiffness of suspender is 0.010–0.21 of that of the steel rod with the same section; the axial tensile stiffness of the conventional suspender is basically the same as that of the steel rod with the same section, but the torsional stiffness is only 0.012–0.023 of the steel rod with the same section, and the tension-torsion coupling effect cannot be ignored. The research results provide a concise method to calculate the axial response of semi-parallel wire suspenders, which lays a foundation for the engineering treatment of the untwisting problem of semi-parallel wire suspender using spherical plain bearings.

A CPC fractional model of the heat and mass transport mechanism in Carbon nanotubes with slip effects on velocity

A fractional technique is used to evaluate the temperature, mass, and velocity flow of single and double wall CNTs over a vertical plate. Slip boundary conditions and applied magnetic force are addressed. Human blood is used to examine how base fluid behaves. Applying the proper dimensionless variables results in the dimensionless formulation of initial and boundary conditions related to the governed dimensional concentration, momentum, and energy equations. The Laplace transform technique is used to resolve the dimensionless governing partial differential equations and get the solutions. The constant proportional Caputo (CPC) time-fractional derivative is a unique class of fractional model used in the simulation technique. The fundamental definitions are used to support the said model first. Using MWCNTs and SWCNTs in comparison to the flow characteristics, a thermal and mass study is given. The heat and mass transfer processes for single-walled carbon nanotubes (SWCNTs) have been shown to typically be progressive. The momentum profile decreases as the fractional variables rise. Multi-walled carbon nanotubes (MWCNTs) show more progressive velocity control as a result of the magnetic parameter. Graphs demonstrate the influence of embedding factors on the velocity, energy, and concentration profiles.

A 2D Hydraulic Simulation Model Including Dynamic Piping and Overtopping Dambreach

Numerical simulation of unsteady free surface flows using depth averaged equations that consider the presence of initial discontinuities has been often reported for situations dealing with dam break flow. The usual approach is to assume a sudden removal of the gate at the dam location. Additionally, in order to prevent any kind of dam risk in earthen dams, it is very interesting to include the possibility of a progressive dam breach leading to dam overtopping or dam piping so that predictive hydraulic models benefit the global analysis of the water flow. On the other hand, when considering a realistic large domain with complex topography, a fine spatial discretization is mandatory. Hence, the number of grid cells is usually very large and, therefore, it is necessary to use parallelization techniques for the calculation, with the use of Graphic Processing Units (GPU) being one of the most efficient, due to the leveraging of thousands of processors within a single device. The aim of the present work is to describe an efficient GPU-based 2D shallow water flow solver (RiverFlow2D-GPU) supplied with the formulation of internal boundary conditions to represent dynamic dam failure processes. The results obtained indicate that it is able to develop a transient flow analysis taking into account several scenarios. The efficiency of the model is proven in two complex domains, leading to >76× faster simulations compared with the traditional CPU computation.

Dirichlet and Neumann boundary conditions for a lattice Boltzmann scheme for linear elastic solids on arbitrary domains

We propose novel, second-order accurate boundary formulations of Dirichlet and Neumann boundary conditions for arbitrary curved boundaries, within the scope of our recently introduced lattice Boltzmann method for linear elasticity. The proposed methodology systematically constructs and analyzes the boundary formulations on the basis of the asymptotic expansion technique; thus, it is expected to be of general applicability to boundary conditions for lattice Boltzmann formulations, well beyond the focus of this paper. For Dirichlet boundary conditions, a modified version of the bounce-back method is developed, whereas the Neumann boundary conditions are based on a novel generalized ansatz proposed in this work. The Neumann boundary formulation requires information from one additional neighbor node, for which a finite difference algorithm is introduced that enables the handling of very general boundary geometries. The theoretical derivations are verified by convergence studies using manufactured solutions, as well as by comparison with the analytical solution of the classical plate with hole problem.

Evaluation of Accelerator Pedal Strength under Critical Loads Using the Finite Element Method

The core idea of the research consists in a formulation of boundary conditions of a mechanical accelerator pedal’s strength in an Ansys environment, whose conditions are equivalent to full-scale tests under the critical loads defined by the UNECE’s Regulation No. 13. The lack of regulatory requirements for the strength of pedal types other than brake pedals is a major gap in vehicle certification, especially when it comes to agricultural machinery. In such cases, the authors suggest being guided by UNECE R 13 regarding the strength of the accelerator and other types of pedals and checking their behavior under loads of at least 1000 N. The real value of the yield strength of the material (Silumin 4000) is very important, both in the physical real-life experiments and in FEA simulation. The critical case of a short-term shock loading of the pedal in its extreme position has been considered separately. With the help of the Ansys Explicit Dynamics module, results of a pedal’s behavior were obtained; it lost its integrity and suffered destruction. It is also necessary to check the intermediate stress values depending on the loads for direct and hybrid tasks using the Transient Structural module in order to estimate other critical cases of the pedal behavior.

Numerical Analysis of Unsteady Heat Transfer in the Chamber in the Piston Hybrid Compressor with Regenerative Heat Exchange

The adjoint thermodynamic and heat exchange processes in the new class piston compressor with regenerative heat exchange are considered in the paper. The implicit tridiagonal matrix algorithm is implemented to study the unsteady thermal conductivity in the cylinder–piston group. After the formulation of proper initial and boundary conditions, the optimal relationship between temporal and spatial discretization steps is determined. Two different time steps are used in the numerical solution of the two-way coupled fast thermodynamic and slow heat exchange models. The relationship between those time steps is determined as well. The conducted numerical experiments allow the analysis of the dynamics of heat exchange in the cylinder–piston group, temperature variation in different parts of the cylinder–piston group, the impact of the heat transfer processes on isothermal and adiabatic efficiency, the impact of heat exchange dynamics on the thermodynamic cycle, as well as other thermodynamic and energetic effects.

Effect of boundary conditions on particle spurious movement in Smoothed Particle Hydrodynamics method

SPH method due to its Lagrangian particle nature suffers from non-physical particle motion resulting from its numerical properties. We study these effects associated with solid walls realization.We used stationary test case to investigate behavior of multiple formulations of boundary conditions for solid walls. Multi-layer boundaries with ghost particles are considered (Dynamic Boundary Conditions, Modified Dynamic Boundary Conditions) as well as one-layer boundary conditions based on discretization of boundary terms and usage of boundary integrals. Some aspects of each realization with respect to the stationary case and particle spurious motion are discussed. Moreover we examined the levels of spurious motion caused by different sources as the free surface, straight walls, sharp corners, … The results show that for the boundary conditions good enough the most significant portion of spurious motion arises in place where the water surface connects to the wall and the spurious motion caused by this interaction is several orders of magnitude higher, then artificial motion caused by other sources.

LaBCof: Lattice Boltzmann boundary condition framework

The boundary condition is one of the most challenging aspects of lattice Boltzmann simulations. One of the reasons that make it difficult to implement a new boundary condition is the dependency of formulation on the orientation of the boundary nodes. So, any boundary condition formulation should be derived, implemented, and verified in terms of the boundary orientation. In this study a new framework for the implementation of boundary conditions in the lattice Boltzmann method has been developed. The main goal of this framework is to implement a boundary condition for one side/corner of a domain which can then be applied to any other side/corner. Since implementing a boundary condition depends on the orientation of the boundary nodes, using this framework makes it easier to implement any set of boundary conditions. The framework is developed in C++ using object-oriented programming. It is then verified by implementing some common boundary conditions in LBM. This framework can be used for any kind of data structure of lattice Boltzmann code.

A case study on the migration of nuclides in the coupled vadose zone-groundwater system at a proposed spent fuel reprocessing site in Gansu, China

Nuclear energy represents a relatively novel clean energy source, and potential nuclear contamination must be considered while developing the nuclear industry. In the context of a hypothetical nuclear accident, a nuclide migration model was established based on regional geological and hydrological data to investigate the migration characteristics of the radionuclides 3H, 99Tc(VII), 137Cs, and 79Se(IV) in a proposed spent fuel reprocessing site (SFRS) in Gansu, China. The temporal and spatial evolution of these four nuclides as well as their effects on the environment and local population were discussed in detail. The model of nuclide migration in the vadose zone was coupled to a large-scale regional nuclide migration model of groundwater in the form of boundary conditions. The results under the least favorable conditions showed that nuclides with high adsorption rates (137Cs and 79Se(IV)) remained in the vadose zone and decayed out completely. In contrast, nuclides with low adsorption rates (3H and 99Tc(VII)) penetrated the vadose zone and polluted the groundwater. The vadose zone delayed the arrival of 3H and 99Tc(VII) to the phreatic surface and reduced their concentrations. It was predicted that the concentration of 3H in the groundwater would fall below the threshold after 40 years, and the distance from the leakage area to the leading edge of the contamination plume would be 374 m. 99Tc(VII), which has a slightly higher adsorption rate and a lower decay rate than 3H, spread 833 m after 3000 years when its concentration was below the threshold. 3H and 99Tc(VII) did not migrate to downstream farms to threaten the safety of residential drinking water. This research presents a safety assessment of a nuclear area in Gansu, China, and provides a significant reference for the future construction of nuclear facilities and nuclear contamination control.

Bilateral XVA pricing under stochastic default intensity: PDE modelling and computation

Adjusting derivative prices to take into account default risk has attracted the attention of several researchers and practitioners, especially after the 2007-2008 financial crisis. We derive a novel partial differential equation (PDE) model for derivative pricing including the adjustment for default risk, assuming that the default risk of one of the counterparties (the buyer) follows a Cox-Ingersoll-Ross (CIR) process, while the other party has constant default risk. The time-dependent PDE derived is of Black-Scholes type and involves two “space” variables, namely the asset price and the buyer default intensity, as well as a nonlinear source term. We formulate boundary conditions appropriate for the default intensity variable. The numerical solution of the PDE is based on standard finite differences, and a penalty-like iteration for handling the nonlinearity. We also develop and analyze a novel asymptotic approximation formula for the adjusted price of derivatives, resulting in a very efficient, accurate, and convenient for practitioners formula. We present numerical results that indicate stable second order convergence for the 2D PDE solution in terms of the discretization size. We compare the effectiveness of the 2D PDE and asymptotic approximations. We study the effect of various numerical and market parameters to the values of the adjusted prices and to the accuracy of the computed solutions.

To E or Not to E : Numerical Nuances of Global Coronal Models

In recent years, global coronal models have experienced an ongoing increase in popularity as tools for forecasting solar weather. Within the domain of up to 21.5 R ⊙, magnetohydrodynamics (MHD) is used to resolve the coronal structure using magnetograms as inputs at the solar surface. Ideally, these computations would be repeated with every update of the solar magnetogram so that they could be used in the ESA Modeling and Data Analysis Working Group magnetic connectivity tool (http://connect-tool.irap.omp.eu/). Thus, it is crucial that these results are both accurate and efficient. While much work has been published showing the results of these models in comparison with observations, not much of it discusses the intricate numerical adjustments required to achieve these results. These range from the details of boundary condition formulations to adjustments as large as enforcing parallelism between the magnetic field and velocity. By omitting the electric field in ideal MHD, the description of the physics can be insufficient and may lead to excessive diffusion and incorrect profiles. We formulate inner boundary conditions that, along with other techniques, reduce artificial electric field generation. Moreover, we investigate how different outer boundary condition formulations and grid design affect the results and convergence, with special focus on the density and radial component of the B -field. The significant improvement in accuracy of real magnetic map–driven simulations is illustrated for an example of the 2008 eclipse.