Discretisation of the velocity-pressure formulation with integrated radial-basis-function networks

This paper presents a preconditioning scheme to improve the condition number of integrated radial-basis-function (RBF) matrices in solving large-scale 2D elliptic problems. The problem domain is discretised using a Cartesian grid, over which integrated RBF networks are employed to represent the field variable. The present preconditioner is constructed from 1D integrated RBF networks along grid lines. Test problems defined on rectangular and non-rectangular domains are employed to study the performance of the scheme.

A novel approximation method using integrated radial basis function networks (IRBFN) coupled with moving least square (MLS) approximants, namely moving integrated radial basis function networks (MIRBFN), is proposed in this work. In this method, the computational domain w is divided into finite sub-domains w which satisfy point-wise overlap condition. The local function interpolation is constructed by using IRBFN supported by all nodes in subdomain w. The global function is then constructed by using Partition of Unity Method (PUM), where MLS functions play the role of partition of unity. As a result, the proposed method is locally supported and yields sparse and banded interpolation matrices. The computational efficiency are excellently improved in comparison with that of the original global IRBFN method. In addition, the present method possesses the Kronecker-d property, which makes it easy to impose the essential boundary conditions. The proposed method is applicable to randomly distributed datasets and arbitrary domains. In this work, the MIRBFN method is implemented in the collocation of a first-order system formulation to solve PDEs governing various problems including heat transfer, elasticity of both compressible and incompressible materials, and linear static crack problems. The numerical results show that the present method offers high order of convergence and accuracy.

This work deals with the benchmarking of newly implemented thermal boundary conditions for the incompressible Boussinesq heat transfer solver in OpenFOAM. Internal fluid flows for building physics purposes can be accurately modelled with the incompressible Boussinesq approach. In order to accurately model thermal heat transfer phenomena using CFD more flexible Neumann and Robin thermal boundary condition are required which do not exist in the standard package of OpenFOAM for incompressible Boussinesq solvers. A new boundary condition is derived from other compressible solvers in OpenFOAM and adapted for use with the Boussinesq approximation. This new boundary condition is benchmarked against the commercial software ANSYS CFX by means of an assessment of the mean air temperature and the mean velocities as well as the mean radiant temperature. A simple L-shaped room case is used to compare results between OpenFOAM and ANSYS CFX. The mesh used is for both softwares identical. Six different scenarios are investigated by varying the adjacent wall temperatures (Robin boundary condition). For one scenario, the Neumann boundary condition is also applied. Furthermore, radiation coupling is integrated into the new boundary condition and included in the model. The radiation models view factors (viewFactor) and discrete ordinate method (fvDOM) in OpenFOAM are compared. The comparison the two softwares show very good agreement for air and mean radiant temperatures with a maximum of 0.25 K deviation. However, the CFX results show consistently a more pronounced fluid flow within the room. The results between fvDOM and viewFactor within OpenFOAM are quite similar to each other and furthermore match well with the results obtained in CFX.

This paper presents the development of a mathematical model of non-isothermal, unsteady flow of a real gas in deviation of Darcy's law and an algorithm for its numerical implementation. A formula for the flow velocity for non-linear binomial law of filtration was obtained. The formula includes a radial coordinate pressure derivative, thermo physical gas properties, and geophysical rock properties. There is also a partial differential equation and boundary condition at the well bottomhole for non-linear two-dimensional law of filtration. A method for simultaneous numerical integration for finite differences of differential equations shows an unsteady non-isothermic gas flow in the reservoir and energy conservation equation with initial and boundary conditions on the well bottom and the external drainage radius; functions of pressure and temperature at different time variances and their distribution along radial coordinates were calculated using a developed mathematical model. Graphs for these functions for linear and non-linear flow in reservoirs with different reservoir features for different well rates were built. Analysis of these graphs shows that taking into account the deviation of Darcy’s law of gas filtration in high permeable reservoirs for various flow rates does not make substantial corrections to the calculated values of pressure and temperature in comparison with the corresponding characteristics for flow. Calculations of bottomhole pressure - assuming that gas flow is steady - have calculation errors that are less than 0.2%, considered in the numerical implementation of task solution. With a low-permeable reservoir - when well production rates are low (for example, 169 thousand sm 3/day) - the bottomhole pressure decrease is small; therefore, a decrease of temperature at the bottomhole zone will also be small. The difference between the calculated values of these characteristics for linear and nonlinear filtering laws will also be negligible. Error of the value of bottomhole pressure calculated in assumptions that flow is steady does not exceed 0.7%. There is a deviation of Darcy's law in gas flow through the low-permeable reservoir at high well production flow rates (for example, 845 thousand sm 3/day), which leads to a substantial reduction of the reservoir pressure and temperature. Pressure stabilization for gas flow in the low-permeable reservoir does not depend on flow rate value. Error of value of bottomhole pressure, calculated in assumption that flow is steady exceeds 300%.

This paper explores the application of the mesh-free Local Radial Basis Function Collocation Method (LRBFCM) [1] in solution of coupled heat transfer and fluid flow problems associated with solidification of a binary alloy. The involved temperature, velocity, species and pressure fields are represented on overlapping sub-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBF’s. The energy and momentum equations are solved through explicit time stepping. The pressure-velocity coupling is calculated iteratively, with pressure correction, predicted from the local continuity equation violation [2,3]. The solution procedure is demonstrated in simulation of solidification of 8 wt % Pb-Sn alloy and 5 wt % Sn-Pb alloy in rectangular cavity at conditions, defined in [4]. The numerical examples include studies with different uniform discretization and its influence on final macrosegregation map. The solution is assessed by comparison with the reference results of the finite volume and finite element method. The advantages of the method are simplicity, accuracy and straightforward applicability in non-uniform node arrangements. The methodology will be use in complementing our heat transfer model [5] of the continuous casting of steel.

In the literature immersed boundary methods are employed tosimulate complex flows around moving arbitrary bodies without thenecessity of remeshing. These methods employ a regular Eulerian meshto simulate the fluid flow and a Lagrangian representation of theboundary of the bodies. The two representations can be coupled throughan immersed boundary condition constraining the fluid to exactlyfollow the boundary of the bodies (immersed boundaries). Typicallysuch methods suffer from accuracy problems, that arise from spuriousmass fluxes over the immersed boundary (IB), pressure boundaryconditions or high density ratios. The mirroring IB method Mark(2008); Mark and van Wachem (2008) resolves these problems by ensuringzero mass flux over the IB instead of employing a pressure boundarycondition. In this work the mirroring IB method together with a hybridIB condition are implemented and validated in IBOFLOW. IBOFLOW is anincompressible finite-volume based fluid flow solver. TheNavier-Stokes' equations are coupled with the SIMPLEC Doormaal andRaithby (1984) method and discretized on a Cartesian octree grid thatcan be dynamically refined and coarsened, enabling grid refinement tofollow moving bodies. The variables are stored in a co-locatedconfiguration and pressure weighted flux interpolation Rhie and Chow(1983) is employed to prevent pressure oscillations. In theimplemented IB method the immersed bodies are represented by ananalytical description or by a triangulation. The method models thepresence of the bodies inside the fluid by an implicitly formulated IBcondition, which constrains the fluid velocity to the boundaryvelocity with second-order accuracy. The original mirroring IBcondition mirrors the velocity field over the local IB and the hybridIB condition mirrors and extrapolates the fluid velocity onto the IB.These IB conditions generate a fictitious velocity field inside thebodies, which is excluded in the continuity equation to ensure zeromass flux over the boundary. The fluid flow over an immersed sphere issimulated to validate and compare the different IB conditions. Thesimulated drag force is compared to experimental findings withexcellent agreement and a detailed convergence study of the error ofthe fluid velocity integrated over the immersed boundary is performedto show the strictly second-order accuracy of the implemented IBconditions. It is shown that the error is reduced with the hybrid IBcondition compared to the original mirroring IB condition. Inaddition, a sedimenting sphere with a moving grid refinement issimulated to validate the hybrid method and show the potential of thedynamic octree grid.

For n > 1, we consider a reaction-diffusion equationut = Δu + α(y)∇ · G(u) + f(u), (0.2)in an unbounded cylinder Ω := R×D, where D ⊂ Rn−1 is a smooth bounded domain, with a presence of a convection term, under both Neumann and Dirichlet boundary conditions on ∂Ω. For both types of boundary condition, we consider two different forms of convection term, namely : α(y)∇·G(u) and∇ · (α(y)G(u)). The reaction term f is “monostable”. In both Neumann and Dirichlet cases, we prove that there exists a critical speed c⋆ ∈ R such that there exists a travelling front solution of the form u(x, t) = w(x1 −ct, y) with speed c if and only if c ≥ c⋆, where x1 is the coordinate corresponding to theaxis of the cylinder. The critical speed c⋆ often plays an important role for monostable problems by characterizing the long-time behaviour of the initial value problem. The existence of travelling waves for all c ≥ c⋆ is typical of monostable problems such as the prototype Fisher-KPP equation.We give a min-max formula for the speed c⋆. For both types of boundary conditions, we prove that c⋆ is bounded below by a quantity c′ which is related to a certain eigenvalue problem, associated with the linearized problem around 0. Note that under Dirichlet boundary conditions, an extra assumption is needed to ensure that c′ exists, namely, f′(0) has to be greater than the principal eigenvalue of the linearized operator. We discuss two special cases where the equality c⋆ = c′ holds. Under both Neumann and Dirichlet boundary conditions, the first special case is when G = (G1, 0, · · ·, 0), assuming the so-called KPP condition for f and that α(y)G′ 1(u) ≥ α(y)G′ 1(0), for all y ∈ D and all u ∈ (0, 1). The second case is treated only under Neumann boundary conditions : when G′ 1(0) = 0, assuming the KPP condition for f, and that α(y)G′ 1(u) ≥ 0, for all y ∈ D and u ∈ (0, 1). Note that in that case, we give an explicit formula : c⋆ = c′ = 2 p f′(0). Under Dirichlet boundary conditions, we highlight the influence of the domain D, the reaction term f and the convection term α(y)∇ · G(u) on the critical speed c⋆. In the special case where G = (G1, 0, ···, 0), using that c⋆ = c′, we use the eigenvalue problem related to c′ to establish some optimization results for c⋆.

The nonlinear Schrödinger equation in the finite line

Imposition of boundary conditions for elliptic equations in the context of non boundary fitted meshless methods

Fujii, Mimura, and Nishiura [1985] and Armbruster and Dangelmayr [1986, 1987] have observed that reaction-diffusion equations on the interval with Neumann boundary conditions can be viewed as restrictions of similar problems with periodic boundary conditions; and that this extension reveals the presence of additonal symmetry constraints which affect the generic bifurcation phenomena. We show that, more generally, similar observations hold for multi-dimensional rectangular domains with either Neumann or Dirichlet boundary conditions, and analyse the group-theoretic restrictions that this structure imposes upon bifurcations. We discuss a number of examples of these phenomena that arise in applications, including the Taylor-Couette experiment, Rayleigh-Benard convection, and the Faraday experiment.

We announce a new result which shows that under either Dirichlet, Neumann, or Robin boundary conditions, the corners in a planar domain are a spectral invariant of the Laplacian. For the case of polygonal domains, we show how a locality principle, in the spirit of Kac's "principle of not feeling the boundary" can be used together with calculations of explicit model heat kernels to prove the result. In the process, we prove this locality principle for all three boundary conditions. Albeit previously known for Dirichlet boundary conditions, this appears to be new for Robin and Neumann boundary conditions, in the generality presented here. For the case of curvilinear polygons, we describe how the same arguments using the locality principle fail, but can nonetheless be replaced by powerful microlocal analysis methods.

Following the inoculation of stripe rust disease of different levels in the winter wheat field, thes measured the stripe rust disease severity and winter wheat canopy reflectance at different growth stages. PLS (Partial Least Square) was adopted to build a regression model for disease severity inversion from canopy reflectance. The results indicate that the inversion accuracy of PLS is higher than that of the method proposed in reference[4], which used PRI (Photochemical Reflectance Index) to predict disease severity. Regression coefficients of PLS were investigated to obtain useful knowledge. It has been found that the first derivatives on the two sides of the chlorophyll absorption valley (505～550 nm,640～670 nm,680～700 nm) are most important in determining disease severity. Diseased winter wheat has higher absolute values of first derivatives in the three spectral regions.

A new method for numerical representation of the land-water boundary in lake circulation models

Our purpose is to establish a numerical method meeting the requirements of efficiency, stability, accuracy and easy-using. Faced with these demands, in this paper, a hybrid method combining the advantages of 2 weak-form meshless methods is proposed for solving steady and transient heat conduction problems with temperature-dependent thermophysical properties in heterogeneous media. In the process of the calculation for an arbitrary model discretized by the hybrid method, two different weak-form methods are used for the nodes categorized as: (1) boundary nodes, (2) near boundary interior nodes and (3) interior nodes. A Galerkin free element method (GFREM) is proposed for the interior nodes of the Cartesian grid; and the local radial point interpolation method (LRPIM) is used for the other two types of nodes. The GFREM using the Cartesian grid can greatly improve the accuracy and efficiency of the hybrid scheme. Special decomposition technology to determine the irregular integral domain for LRPIM can increase the flexibility for complex models. In addition, the full implicit finite difference scheme and Newton–Raphson method are used for solving transient nonlinear problems. And non-crossing interface integral domain and support domain are employed for heterogeneous media. Compared with other methods, numerical results show better accuracy, stability and efficiency from the numerical experiments.

This paper aims at presenting a numerical method for estimating the elastic modulus of recycled concrete with crushed aggregates. In the method, polygonal aggregates following a given sieve curve are generated, and placed into a square simulation element with the aid of the periodic boundary condition and the overlap criterion of two polygonal aggregates. The mesostructure of recycled concrete is reconstructed by embedding an old interfacial transition zone (ITZ) layer inside each recycled aggregate and by coating all the aggregates with a new ITZ layer. The square simulation element is discretized into a regular grid and a representative point is selected from each sub-element. The iterative method is combined with the fast Fourier transform to evaluate the elastic modulus of recycled concrete. After the validity of the numerical method is verified with experimental results, a sensitivity analysis is conducted to evaluate the effects of key factors on the elastic modulus of recycled concrete. Numerical results show that the elastic modulus of recycled concrete increases with the increase of the total aggregate content and the elastic moduli of old and new ITZ but decreases with increasing the replacement ratio of recycled aggregate and the thicknesses of old and new ITZ. It is also shown that, for a replacement ratio of recycled aggregate smaller than 0.3, the elastic modulus of recycled concrete is reduced by no more than 10%.