Imposed Boundary Conditions Research Articles

NUMERICAL AND EXPERIMENTAL INVESTIGATION OF A MICRO GAS TURBINE COMBUSTION CHAMBER

Micro gas - turbines (MGT) offer many advantages such as higher thermal efficiency and reduced noise, and are suitable sources for power generation due to their fuel flexibility, small sizes, and high efficiencies. In recent years, there has been an increase interest in developing MGT for transportation platforms such as Range Extender for Electric Vehicle (REEV), Unmanned Ground/Air Vehicles (UGV/UAV), Auxiliary Power Units (APU). For these applications, the MGT must meet essential requirements like reliability, reasonable price, ecological safety, low noise and vibration, multi-fuel, etc. This paper presents the numerical and experimental investigation of a newly designed annular type combustion chamber. This combustion chamber is part of a 40 daN micro gas turbine, destined to equip a small-scale multifunctional airplane. The combustion chamber is equipped with six innovative vaporizers, using Jet-A as fuel, patented by INCDT COMOTI. The experimental installation on which the combustion tests have been performed consists of: the fuel supply system, an air source, the combustion chamber assembly, a chimney for flue gas exhaust. During the combustion chamber testing campaign, the following parameters have been monitored and registered: air mass flow, air temperature, and pressure before the combustion chamber entrance, the temperature at the combustion chamber exit, the temperature before the pressure regulating valve placed on the exhaust pipe. After the testing campaign has been concluded the numerical simulations have been resumed. A three-dimensional RANS numerical integration of the Navier-Stokes equations has been carried out, using an Eddy Dissipation Combustion Model (EDM) and the k-ε turbulence model, implemented in a numerical simulation conducted using the commercial software ANSYS CFX. The computational domain has been modified in order to match the testing rig. Due to the complex geometry of the computational domain, an unstructured type computational grid has been used. The imposed boundary conditions have been changed in order to match the testing conditions and functioning regimes. A kerosene – air two steps reaction mechanism, with NO formation, has been used. The numerical simulation results have been compared with the parameters measured experimentally, thus validating the obtained results.

Kibble−Zurek scaling of the dynamical localization−skin effect phase transition in a non-Hermitian quasi-periodic system under the open boundary condition

In the present study, the driven dynamics in a non-Hermitian Aubry–André (AA) model under the open boundary condition (OBC) are studied. For this model, non-Hermiticity is introduced by the non-reciprocal hopping, and this model undergoes a localization–skin effect phase transition depending on the strength of the quasi-periodic potential. Although the properties of non-Hermitian systems are very sensitive to the imposed boundary conditions, we find that the scaling behavior can also be described by the same set of the exponents under the periodic boundary condition (PBC). When the initial state is prepared deep in the localized phase and the potential strength is slowly driven through the critical point, we find that the driven dynamics of the localization length ξ and the inverse participation ratio (IPR) could be described by the Kibble–Zurek scaling (KZS). Then, we numerically verify these predictions for different initial states. Finally, the dynamical emergence of the skin effect state is found, and the dynamics can also be described by the Kibble−Zurek scaling with the same set of critical exponents.

Observation of size-dependent boundary effects in non-Hermitian electric circuits

The non-Hermitian systems with the non-Hermitian skin effect (NHSE) are very sensitive to the imposed boundary conditions and lattice sizes, which lead to size-dependent non-Hermitian skin effects. Here, we report the experimental observation of NHSE with different boundary conditions and different lattice sizes in the unidirectional hopping model based on a circuit platform. The circuit admittance spectra and corresponding eigenstates are very sensitive to the presence of the boundary. Meanwhile, our experimental results show how the lattice sizes and boundary terms together affect the strength of NHSE. Therefore, our electric circuit provides a good platform to observe size-dependent boundary effects in non-Hermitian systems.

The imposition of nonconforming Neumann boundary condition in the material point method without boundary representation

Since its original formulation, the material point method (MPM) has been repeatedly further developed to become a powerful tool for simulating large deformation problems. However, the imposition of boundary condition in the MPM remains a challenge since most material points are not located at the boundary and regular background mesh is most commonly used. Therefore, the existing methods to impose boundary condition always require a boundary tracking or reconstruction algorithm, which is not always efficient or accurate. In the current work, a novel strategy to apply nonconforming Neumann boundary condition is proposed that does not require the exact boundary position. The original problem with a nonconforming traction boundary condition is transformed into an equivalent problem with a prescribed virtual stress field. Importantly, both the original and the transformed problems produce the exact same response within the material domain. To solve the equivalent problem, a modified governing equation without boundary representation is subsequently constructed, where the volume integral terms are computed by both particle quadrature and cell quadrature. Several numerical examples are investigated to assess the accuracy and demonstrate the capability of the proposed approach in simulating different engineering problems. These numerical examples include mesh refinement to illustrate the method’s good convergence and a 3D complex surface to illustrate the method’s capability.

Unidirectionality and Husimi functions in constant-width neutrino billiards

We investigate the spectral properties and Husimi functions of relativistic quantum billiards (QBs) consisting of a spin-1/2 particle governed by the Dirac equation and confined to a planar domain of constant-width (CW) by imposing boundary conditions (BCs) on the spinor components. We consider those of neutrino billiards (NBs) proposed in (Berry and Mondragon 1987 Proc. R. Soc. A 412 53). The classical dynamics of billiards of corresponding shape is predominantly chaotic. CW billiards attracted particular attention because they exhibit unusual properties. Their classical dynamics features unidirectionality, whereas in the corresponding nonrelativistic QB a change of the rotational direction of motion is possible via dynamical tunneling, and the spectral properties coincide with those of typical quantum systems with violated time-reversal invariance. Unidirectionality of the quantum dynamics would arise in the structure of the Husimi functions. We analyze them for two realizations of CW NBs in the ultra-relativistic, i.e. the massless case and for massive cases and come to the result, that the modes can be separated into clockwise and counterclockwise modes and dynamical tunneling is absent. This is attributed to the BCs and the unidrectionality of the local current arising from them.

Neutrally stable double-curved shells by inflection point propagation

Elastic structures that can deflect without springback, known as neutrally stable structures, form a remarkable group within their field, since they require the energetic state to remain unchanged during elastic deformation. Several examples in the literature obtain this state of neutral stability by the application of pre-stress, either as a result of manufacturing processes or the application of imposed boundary conditions. In this paper, we present a new class of neutrally stable structure that exhibits neutral stability as part of a continuous deformation process, while also allowing a stress-free configuration to exist. The transition of a double-curved compliant shell from its stress-free stable equilibrium towards its second stable equilibrium, through a range of neutrally stable equilibrium configurations forms the basis of this investigation. To design this neutrally stable shell, an optimization is employed to obtain an ideal set of variables that defines a varying thickness profile. Numerical analysis of the resulting optimized shell structure predicts a substantial region of near-constant energy and associated near-zero loads within this unique deformation mode. Additively manufactured prototypes demonstrate the validity of the modeled results by featuring a continuous equilibrium within the range of motion. These results lay the foundation for compliant beam elements with a neutrally stable bending degree of freedom.

A third-order weighted nonlinear scheme for hyperbolic conservation laws with inverse Lax-Wendroff boundary treatment

Cartesian grids are often used in applications due to the simplicity of grid generation and the efficiency of discretization algorithms. However, for problems with curved boundaries, grid points are often away from the boundaries, leading to the issue of imposing boundary conditions. Recently, it was shown that the inverse Lax-Wendroff (ILW) boundary treatment is very useful for addressing this issue. While most of the investigations focus on the WENO schemes for hyperbolic conservation laws, we present in this paper a new third-order weighted nonlinear scheme, which is based on the framework of weighted compact nonlinear schemes. We show that the scheme is applicable directly on Cartesian grids by using the ILW boundary treatment. To demonstrate the linear stability of the scheme, both the method of eigenvalue analysis and the Gustafsson-Kreiss-Sundström (GKS) theory are employed and analyzed in details. Some numerical tests are also performed to show the accuracy and validity of the proposed method.

Parametric analysis of an efficient boundary condition to control outlet flow rates in large arterial networks

Substantial effort is being invested in the creation of a virtual human—a model which will improve our understanding of human physiology and diseases and assist clinicians in the design of personalised medical treatments. A central challenge of achieving blood flow simulations at full-human scale is the development of an efficient and accurate approach to imposing boundary conditions on many outlets. A previous study proposed an efficient method for implementing the two-element Windkessel model to control the flow rate ratios at outlets. Here we clarify the general role of the resistance and capacitance in this approach and conduct a parametric sweep to examine how to choose their values for complex geometries. We show that the error of the flow rate ratios decreases exponentially as the resistance increases. The errors fall below 4% in a simple five-outlets model and 7% in a human artery model comprising ten outlets. Moreover, the flow rate ratios converge faster and suffer from weaker fluctuations as the capacitance decreases. Our findings also establish constraints on the parameters controlling the numerical stability of the simulations. The findings from this work are directly applicable to larger and more complex vascular domains encountered at full-human scale.

Exact Toverline{T} deformation of two-dimensional Yang-Mills theory on the sphere

We study the Toverline{T} deformation of two-dimensional Yang-Mills theory at genus zero by carrying out the analysis at the level of its instanton representation. We first focus on the perturbative sector by considering its power expansion in the deformation parameter μ. By studying the resulting asymptotic series through resurgence theory, we determine the nonperturbative contributions that enter the result for μ < 0. We then extend this analysis to any flux sector by solving the relevant flow equation. Specifically, we impose boundary conditions corresponding to two distinct regimes: the quantum undeformed theory and the semiclassical limit of the deformed theory. The full partition function is obtained as a sum over all magnetic fluxes. For any μ > 0, only a finite portion of the quantum spectrum survives and the partition function reduces to a sum over a finite set of representations. For μ < 0, nonperturbative contributions regularize the partition function through an intriguing mechanism that generates nontrivial subtractions.

Phase-field regularised cohesive zone model for interface modelling

A discrete interface is represented as a smeared interface in the framework of phase field regularisation. Due to the numerical challenge when imposing boundary conditions we prescribe the phase field in the domain using analytical solutions. We obtain the displacement and the strain from the variational form of the energy functional, but also in the framework of the regularised extended finite element method. Indeed, both approaches result in identical forms, validating the rigour of the method. We have examined different forms of the phase field model and conclude that models with a compact support are more appropriate for modelling interfaces. The method combines advantages of discrete and smeared approaches. Compared to discrete interface models, the method holds similar properties as the extended finite element method (XFEM): no need to treat cracks as geometric discontinuities and avoiding mesh refinement around crack tips. Different from the XFEM, however, the method does not introduce enrichment functions to describe cracks. An advantage compared to the phase field method is that this method directly employs the cohesive zone law from the discrete model, which is physically relevant. The accuracy of the approach for cohesive interface modelling is demonstrated by several numerical examples, including a bar, an L-shaped specimen, and a fibre embedded in an epoxy matrix.

Deep learning phase‐field model for brittle fractures

AbstractWe present deep learning phase‐field models for brittle fracture. A variety of physics‐informed neural networks (PINNs) techniques, for example, original PINNs, variational PINNs (VPINNs), and variational energy PINNs (VE‐PINNs) are utilized to solve brittle phase‐field problems. The performance of the different versions is investigated in detail. Also, different ways of imposing boundary conditions are examined and are compared with a self‐adaptive PINNs approach in terms of computational cost. Furthermore, the data‐driven discovery of the phase‐field length scale is examined. Finally, several numerical experiments are conducted to assess the accuracy and the limitations of the discussed deep learning schemes for crack propagation in two dimensions. We show that results can be highly sensitive to parameter choices within the neural network.

MHD flow of time-fractional Casson nanofluid using generalized Fourier and Fick's laws over an inclined channel with applications of gold nanoparticles

Gold nanoparticles are commonly used as a tracer in laboratories. They are biocompatible and can transport heat energy to tumor cells via a variety of clinical techniques. As cancer cells are tiny, properly sized nanoparticles were introduced into the circulation for invasion. As a result, gold nanoparticles are highly effective. Therefore, the current research investigates the magnetohydrodynamic free convection flow of Casson nanofluid in an inclined channel. The blood is considered as a base fluid, and gold nanoparticles are assumed to be uniformly dispersed in it. The above flow regime is formulated in terms of partial differential equations. The system of derived equations with imposed boundary conditions is non-dimensionalized using appropriate dimensionless variables. Fourier's and Fick's laws are used to fractionalize the classical dimensionless model. The Laplace and Fourier sine transformations with a new transformation are used for the closed-form solutions of the considered problem. Finally, the results are expressed in terms of a specific function known as the Mittag-Leffler function. Various figures and tables present the effect of various physical parameters on the achieved results. Graphical results conclude that the fractional Casson fluid model described a more realistic aspect of the fluid velocity profile, temperature, and concentration profile than the classical Casson fluid model. The heat transfer rate and Sherwood number are calculated and presented in tabular form. It is worth noting that increasing the volume percentage of gold nanoparticles from 0 to 0.04 percent resulted in an increase of up to 3.825% in the heat transfer rate.

Electromagnetic flow of casson nanofluid over a vertical riga plate with ramped wall conditions

The present study aims to investigate the Casson nanofluids flow theoretically over a vertical Riga plate. The temperature and concentration with ramped and isothermal wall conditions are considered. Moreover, the fluid is considered electrically conducted. Concrete is considered as a base fluid while clay nanoparticles are added to it. The present flow regime is formulated in terms of a system of partial differential equations. Using dimensionless variables, the system of equations with the imposed boundary conditions is non-dimensionalized. The Laplace transform technique is used to calculate the exact solutions for the temperature, concentration, and velocity distributions. The effect of various embedded parameters on the velocity, temperature, and concentration fields are shown graphically and discussed physically. The variation in the skin friction for various values of clay nanoparticles volume fraction is shown in tabular form. The results indicate that adding 4% clay nanoparticles, enhanced the skin friction up to 7.04% in instance of ramped wall temperature (RWT) and 11.13% in isothermal wall temperature (IWT). This result is worth noting because the increase in skin friction causes rapid compaction of the cementitious materials and improves the tensile strength of the concrete.

Volume penalization method for solving coupled radiative-conductive heat transfer problems in complex geometries

A novel immersed boundary method for simulating coupled radiative-conductive heat transfer based on a volume penalization method is proposed and verified. A smooth source/sink term is introduced to the governing equations for radiative and conductive heat transfer in order to take into account the absorption, reflection and emission at a solid surface. The proposed algorithm is successfully implemented into an open-source CFD solver, OpenFOAM with adaptive mesh refinement to guarantee sufficiently fine grid resolution near solid surfaces. Benchmark problems of pure radiation and combined conduction-radiation are considered for verifying the proposed method. The present simulation results under various absorption and emission conditions show good agreement with analytical solutions as well as existing literature data within around 5% errors. In addition to its superior performance in terms of accuracy, the present approach does not require special treatments around the solid surface with arbitrary complex geometry to impose boundary conditions for radiative heat transfer, and thereby wide applicability in scientific and engineering problems.

Derivation and analysis of a 1D porous medium flow solver embedded in a two-domain model for 2D and 3D transpiration cooling

• A 1D porous medium flow solver is derived and analyzed. • The 1D solver is embedded in a two-domain model for 2D/3D transpiration cooling. • Differences between results of the new model and a full 2D/3D model are negligible. • Transpiration cooling problems are solved significantly faster with the new model. For the numerical investigation of transpiration cooling problems, often two-domain models are used. Different systems of governing equations are solved in a hot gas and a porous medium domain, and both domains are coupled with each other by imposing boundary conditions on the interface. This work presents a novel approach which splits the porous medium domain into 1D problems for Darcy-Forchheimer flow under local thermal nonequilibrium. The fluid and solid temperature solutions are computed analytically, and only one ordinary differential equation has to be solved to determine fluid density and Darcy velocity. Unique solutions to the equations of the 1D model are verified to exist under physically reasonable assumptions. The solutions of the 1D problems are assembled to one 2D or 3D solution. In a postprocessing, slip and adiabatic side walls are modeled despite being nonexistent in 1D. Numerical 2D and 3D test cases together with experimental data verify the suitability of the new assembled-1D approach for the simulation of transpiration-cooled hot gas flows with orthogonal cooling gas injection. Furthermore, the test cases demonstrate the significant computational speed-up related to a full-2D/3D approach.

Simulating laser-fluid coupling and laser-induced cavitation using embedded boundary and level set methods

A computational method for simulating thermal cavitation induced by long-pulsed laser is presented. This method accounts for the absorption of laser light by a liquid, the formation of vapor bubbles due to localized heating, and the dynamics of the bubbles and the surrounding liquid. The physical model combines the Euler equations for a compressible inviscid two-phase fluid flow, a reduced form of the radiative transfer equation for laser radiation, and a local thermodynamic model of vaporization. The Euler equations are solved using the FInite Volume method with Exact two-phase Riemann solvers (FIVER). Following this method, numerical fluxes across phase boundaries are computed by constructing and solving one-dimensional bimaterial Riemann problems. The paper focuses on numerical methods for coupling the laser and fluid governing equations and tracking the vapor bubbles. An embedded boundary finite volume method is proposed to solve the laser radiation equation on the same mesh created for the Euler equations, which usually does not resolve the boundary and propagation directions of the laser beam. To impose boundary conditions, ghost nodes outside the laser domain are populated by mirroring and interpolation techniques. The existence and uniqueness of solution are proved for the two-dimensional case, leveraging the special geometry of the laser domain. The order of accuracy of the method is also proved, and verified using numerical tests. A method of latent heat reservoir is proposed to predict the onset of vaporization, which accounts for the accumulation and release of latent heat. A unique challenge associated with long-pulsed laser is that the dynamics of vapor bubbles is driven not only by the inertia of the bubble nuclei, but also by the continuation of vaporization. In this work, the localized level set method is employed to track the bubble surface, and a method of local correction and reinitialization is proposed to account for continuous phase transitions. Several numerical tests are presented to verify the convergence of these methods. Two realistic simulations of laser-induced cavitation are presented at the end, showing that the computational method is able to capture the key phenomena in these events, including non-spherical bubble expansion, shock waves, and the “Moses effect”.

Boundary Element Methods for Helmholtz Problems With Weakly Imposed Boundary Conditions

We consider boundary element methods where the Calder\'on projector is used for the system matrix and boundary conditions are weakly imposed using a particular variational boundary operator designed using techniques from augmented Lagrangian methods. Regardless of the boundary conditions, both the primal trace variable and the flux are approximated. We focus on the imposition of Dirichlet conditions on the Helmholtz equation, and extend the analysis of the Laplace problem from \emph{Boundary element methods with weakly imposed boundary conditions} to this case. The theory is illustrated by a series of numerical examples.

Interpolating Meshless Methods for 3D Elastic Problems

Interpolating meshless methods can directly impose boundary conditions because of the interpolation property which shows advantages in dealing with problems with boundary conditions. The interpolating element-free Galerkin method (IEFGM), the improved interpolating element-free Galerkin method (IIEFGM), and the radial point interpolation method (RPIM) are applied in this paper to solve the two-dimensional and three-dimensional elastic problems. IEFGM and IIEFGM are two different ways to change the status that the traditional element-free Galerkin method (EFG) does not have the interpolation property. IEFGM uses an improved interpolating moving least-squares (IMLS) method that employed singular weight functions while IIEFGM takes the improved interpolating moving least-squares method based on non-singular weight function. RPIM, one of the most widely used interpolating meshless methods, is compared with IEFGM and IIEFGM in this paper. The numerical results of two-dimensional and three-dimensional elastic problems show that the three types of interpolating meshless methods obtain high precision displacement solutions and stress solutions.

Spectral analysis of matrices resulting from isogeometric immersed methods and trimmed geometries

Dealing with complex and trimmed geometries is one of the current challenges in isogeometric analysis. Coupling the isogeometric paradigm with the framework of immersed boundary methods is an attractive practical strategy. In this paper, we consider general variable-coefficient Poisson problems and propose a new formulation of an immersed Galerkin discretization based on tensor-product cardinal B-splines. The weakly imposed boundary conditions do not require any tuning of user-defined penalty or stabilization parameters, and the system matrices are symmetric for symmetric problems. We analyze the spectral behavior of such matrices and prove that they enjoy an asymptotic spectral distribution when the matrix size tends to infinity. We provide an explicit description of this asymptotic distribution which turns out to have a canonical structure incorporating the coefficients of the differential operator and the discretization technique. In the spirit of the isogeometric paradigm, trimmed (single-patch) geometry maps can be treated by means of the same immersed approach. Therefore, we are able to describe spectral properties of matrices in the context of trimmed geometries as well. The theoretical findings are complemented with a selection of numerical experiments.

Efficient angiography simulation for complex vessels

AbstractAngiography simulation is a critical step in virtual interventional surgery. However, the blood vessels are always complex and irregular, which poses a significant challenge to the accuracy and efficiency of simulation. In this paper, we present a novel method to simulate contrast media propagation, which can efficiently and accurately deal with various complex vascular structures and the coupling between blood and contrast media. Our method represents the vascular structures by signed distance functions and to impose boundary conditions, and then we compute the boundary volume by Gauss–Kronrod quadrature, which yields more accurate results even in complex vessels. Furthermore, we improve the simulation's initialization efficiency and real‐time force computational efficiency by precomputing the boundary volume values throughout the vascular structure and saving them to a volume map, which can be queried very efficiently during runtime. Moreover, we add term to the multiple‐fluid model to handle the vascular viscosity, which achieves a more realistic effect. Experiments show that our method obtains more smooth and accurate particle states, and significantly improves initialization efficiency. Finally, we invited 12 clinicians to evaluate the algorithm and verify the clinical value of our algorithm.