Discretization Procedure Research Articles

Efficient aerodynamic shape optimization through reduced order CFD modeling

Although computational power is increasingly available, high-fidelity simulation based aerodynamic shape optimization is still challenging for industrial applications. To make the simulation based optimization acceptable in the practice of engineering design, a technique combining mesh morphing and reduced order modeling is proposed for efficient aerodynamic optimization based on CFD simulations. The former technique avoids the time-consuming procedure of geometry discretization. And the latter speeds up the procedure of field solution by exploiting pre-computed solution snapshots. To test the efficiency of the proposed method, the windshield of a motorbike is analyzed and optimized. It is shown that even the total number of cells of the mesh is around 0.4 million, the CFD computation and the post-processing of the results can be completed in less than 10 s if the reduced order model is adopted. Running on a personal computer, the generic algorithm is applied to optimize the profile of the windshield. A 8% reduction of the drag coefficient is achieved after 800 queries of the reduced order CFD model and the total CPU time is only around 2 h.

Dependencies Between Histogram Parameters and the Kernel Estimate of the Probability Density of a Multidimensional Random Variable

The dependencies between the sampling intervals of the domain of values of a multidimensional random variable and the blur coefficients of the kernel probability density estimate are determined. The results of an analysis of the asymptotic properties of a multidimensional nonparametric estimate of the probability density of the Rosenblatt–Parzen type and its modifications are investigated. Modification of the kernel probability density estimate is a smoothed multi-dimensional histogram. The formulas for calculating the optimal parameters of the kernel function blur coefficients and the lengths of the sampling intervals of the values of the components of a multidimensional random variable are analyzed. Blur coefficients are presented as the products of an indefinite parameter and the mean square deviations of random components. An indefinite parameter and the number of sampling intervals of multidimensional random variables are found from the condition of minimum mean square deviations of the considered probability density estimates. Based on the results obtained, the relationships between the blur coefficient parameters and the discretization procedure of the range of values of a multidimensional random variable are established. The discovered patterns are characterized by a set of constants. The constants under consideration are represented by the ratio of nonlinear functionals of the probability density and their second derivatives with respect to each random component. The values of the constants are characterized by the type of probability density and are independent of their parameters. From computational experimental data, we establish the relationships between the constants and the product of the counterexcess coefficients with the components of a multidimensional random variable. The results obtained make it possible to quickly determine the lengths of sampling intervals of the values of the components of a multidimensional random variable from the kernel function blur coefficients of a nonparametric probability density estimate.

Curved domain discretization with boundary defined by polar equations

A domain discretisation procedure for a planar curved domain with boundary in polar equations is presented. The curved domain is split into curved triangles and then to a fine mesh of linear triangles in the interior and curved triangles or linear triangles near to the boundary. Later by inserting midside nodes to these triangles 6-node triangles obtained further each one into four triangles. The mesh conformity is preserved by applying similar procedure to every triangle of the domain. This procedure is applied to discretize the star shaped curved domain or cracked convex curved domains into all triangles and then into all quadrilaterals. Thus we generate a triangular and a quadrangular finite element mesh. The refinements to the mesh are obtained by increasing the number of divisions of the boundary curve. This discretization of curved domains will reduce the computational complications in the evaluation of integrals, which has lot of practical applications.

The critical threshold for Bargmann–Fock percolation

In this article, we study the excursions sets $\mathcal{D}_p=f^{-1}([-p,+\infty[)$ where $f$ is a natural real-analytic planar Gaussian field called the Bargmann-Fock field. More precisely, $f$ is the centered Gaussian field on $\mathbb{R}^2$ with covariance $(x,y) \mapsto \exp(-\frac{1}{2}|x-y|^2)$. In~\cite{bg_16}, Beffara and Gayet prove that, if $p \leq 0$, then a.s. $\mathcal{D}_p$ has no unbounded component. We show that conversely, if $p>0$, then a.s. $\mathcal{D}_p$ has a unique unbounded component. As a result, the critical level of this percolation model is $0$. We also prove exponential decay of crossing probabilities under the critical level. To show these results, we develop several tools including a KKL-type result for biased Gaussian vectors (based on the analogous result for product Gaussian vectors by Keller, Mossel and Sen in~\cite{keller2012geometric}) and a sprinkling inspired discretization procedure. These intermediate results hold for more general Gaussian fields, for which we prove a discrete version of our main result.

Two Useful Discrete Distributions to Model Overdispersed Count Data

The methods to obtain discrete analogues of continuous distributions have been widely considered in recent years. In general, the discretization process provides probability mass functions that can be competitive with the traditional model used in the analysis of count data, the Poisson distribution. The discretization procedure also avoids the use of continuous distribution in the analysis of strictly discrete data. In this paper, we seek to introduce two discrete analogues for the Shanker distribution using the method of the infinite series and the method based on the survival function as alternatives to model overdispersed datasets. Despite the difference between discretization methods, the resulting distributions are interchangeable. However, the distribution generated by the method of infinite series method has simpler mathematical expressions for the shape, the generating functions and the central moments. The maximum likelihood theory is considered for estimation and asymptotic inference concerns. A simulation study is carried out in order to evaluate some frequentist properties of the developed methodology. The usefulness of the proposed models is evaluated using real datasets provided by the literature.

Sparsity Preserving Discretization With Error Bounds

Typically when designing distributed controllers it is assumed that the state-space model of the plant consists of sparse matrices. However, in the discrete-time setting, if one begins with a continuous-time model, the discretization process annihilates any sparsity in the model. In this work we propose a discretization procedure that maintains the sparsity of the continuous-time model. We show that this discretization out-performs a simple truncation method in terms of its ability to approximate the “ground truth” model. Leveraging results from numerical analysis we are also be able to upper-bound the error between the dense discretization and our method. Furthermore, we show that in a robust control setting we can design a distributed controller on the approximate (sparse) model that stabilizes the dense model.

Efficient aerodynamic shape optimization through free-form mesh morphing and reduced order CFD modeling

Although computational power is increasingly available, high-fidelity simulation based aerodynamic shape optimization is still challenging for industrial applications. To make the simulation based optimization acceptable in the practice of engineering design, a technique combining mesh morphing and reduced order modeling is proposed for efficient aerodynamic optimization based on CFD simulations. The former technique avoids the time-consuming procedure of geometry discretization. And the latter speeds up the procedure of field solution by combining pre-computed solution snapshots. To test the efficiency of the proposed method, the windshield of a motorbike is analyzed and optimized. It is shown that even the total number of cells of the mesh is around 0.4 million, the CFD computation and the post processing of the results can be completed in less than 10 seconds if the reduced order model is adopted. Running on a personal computer, the generic algorithm is applied to optimize the profile of the windshield. A 8% reduction of the drag coefficient is achieved after 800 queries of the reduced order CFD model and the total CPU time is only around 2 hours.

A 3D discretization procedure for the material point method (MPM)

The land topography of certain mountainous regions can be highly variable. The geometry of the topography may be influenced by geological, seismic, weathering, and other processes. The creation of the full 3D numerical models of these topographies represents a challenge. In this work, we present a methodology that allows to create numerical material point method (MPM) models based on the contour lines of the land. The proposed methodology has three main characteristics: the construction of the MPM discrete model is based only on raster digital elevation model (DEM) data; finite element meshes are not required during the process; and the heterogeneities are defined by the DEM data of each material. The application of the proposed methodology to the Daguangbao landslide which occurred in China, in 2008, proved to be adequate for the creation of models with high topography variations and spatial heterogeneities.

On the Calculation of the Vortex Sheet and Point Vortices Effects at Approximate Solution of the Boundary Integral Equation in 2D Vortex Methods of Computational Hydrodynamics

When simulating incompressible flow around airfoil using vortex methods of computational hydrodynamics, it is necessary to solve the boundary integral equation with respect to the intensity of the vortex sheet at the airfoil surface line. For its approximate solution, the Galerkin method with a piecewise-constant and piecewise-linear representation of the solution on the airfoil surface line can be applied. The coefficients of the system of linear equations, arising from the procedure of discretization of the integral equation, e xpress the influence of the vortex sheet located on the panels of the airfoil surface line on other panels; the components of the right-hand side vector express the influence of point vortices that simulate a vortex wake in the flow domain. For the case of the airfoil surface line approximation with a polyline consisting of rectilinear panels, exact analytical expressions are obtained for the coefficients of the matrix and the right-hand side vector; for the case of taking into account the curvature of the panels, an approach for their approximate calculation is proposed and all the necessary formulae are derived.

An improved FE-Meshfree method for solving steady seepage problems

An improved FE-Meshfree method, formulated as a combination of finite element method (FEM) and radial point interpolation method (RPIM), has been proposed in this paper, aiming for obtaining a more accurate and robust solution for steady seepage problems. Two main aspects are mainly investigated: firstly, the traditional cell-based discretization and assembly procedure as normally seen in FEM is replaced by a support domain based one, which potentially improves the computed results with a higher tolerance with the mesh distortion; secondly, an adaptation procedure of inserting extra field nodes on the boundaries/interfaces is proposed and incorporated, which shows an improved computational accuracy and stability. The superiority of the method is demonstrated via some applications to both confined and unconfined seepage problems.

Isogeometric topology optimization of anisotropic metamaterials for controlling high‐frequency electromagnetic wave

SummaryThis study presents a level set–based topology optimization with isogeometric analysis (IGA) for controlling high‐frequency electromagnetic wave propagation in a domain with periodic microstructures (unit cells). The high‐frequency homogenization method is applied to characterize the macroscopic high‐frequency waves in periodic heterogeneous media whose wavelength is comparative to or smaller than the representative length of a unit cell. B‐spline basis functions are employed for the IGA discretization procedure to improve the performance of electromagnetic wave analysis in a unit cell and topology optimization. Also, to keep the same order of continuity on the periodic boundaries as on other element edges in the domain, we propose the extended domain approach, while incorporating Floquet periodic boundary condition (FPBC). Two types of optimization problems are taken as examples to demonstrate the effectiveness of the proposed method in comparison with the standard finite element analysis (FEA). The optimization results provide optimized topologies of unit cells qualified as anisotropic metamaterials with hyperbolic and bidirectional dispersion properties at the macroscale.

A time domain finite element approach based on time series estimation

This paper details the use of time series for solving finite element equations. It was aimed at eliminating the discretization of time intervals to reduce calculation time and to obtain accurate results using time-dependent variables. For this purpose, the matrix form of the time series finite element formulation for both continuous and discontinuous time functions were extracted using Taylor and Fourier series for estimating time-dependent functions. Based on the extracted formulation, for solving elastodynamic problems, a computer code and algorithm was introduced. The efficiency and accuracy of the formulation was evaluated by solving different analytical examples. To reveal the power of time series for modeling real problems three practical examples such as settlement, flood protector structures and temporary cofferdams were modeled using time series approach and the results of analyses were presented. The results showed that having removed time discretization procedure, the time series based finite element method can be efficiently applied to decrease the calculation time and computational efforts. In addition, time-continuous responses resulting from this method, help reduce the calculation steps which is vital in many engineering problems.

Port-Hamiltonian formulation and symplectic discretization of plate models Part II: Kirchhoff model for thin plates

The mechanical model of a thin plate with boundary control and observation is presented as a port-Hamiltonian system (pHs), both in vectorial and tensorial forms: the Kirchhoff-Love model of a plate is described by using a Stokes-Dirac structure and this represents a novelty with respect to the existing literature. This formulation is carried out both in vectorial and tensorial forms. Thanks to tensorial calculus, this model is found to mimic the interconnection structure of its one-dimensional counterpart, i.e. the Euler-Bernoulli beam. The Partitioned Finite Element Method (PFEM) is then extended to obtain a suitable, i.e. structure-preserving, weak form. The discretization procedure, performed on the vectorial formulation, leads to a finite-dimensional port-Hamiltonian system. This part II of the companion paper extends part I, dedicated to the Mindlin model for thick plates. The thin plate model comes along with additional difficulties, because of the higher order of the differential operator under consideration.

Formation of mathematical models of stationary dynamics of structures which include thin elastic plates with distributed inertial parameters

This paper is devoted to the problems of modeling stationary oscillatory processes in elastic systems that contain thin-walled plates. The described method allows us to include the mathematical formalization of thin-walled plates in oscillatory models of elastic structures with irregular boundaries of computational areas and heterogeneous boundary conditions. The formation of dynamic models of this kind is based on the use of discretization methods, in particular, finite element methods that allows approximating the initial models with distributed inertial and rigid parameters — discrete ones with concentration of parameters at certain points — nodes of a dynamic system. Obviously, such approximations are accompanied by errors, the estimation of which is extremely difficult to perform under the conditions of the mentioned heterogeneities and irregularities even in the interval variants. In the present work, the construction of elements of bending stationary vibrations of thin-walled plates under monoharmonic effects is proposed. Possessing all the properties of a finite element, the proposed element contains the values of the distributed masses as parameters. In contrast to the use of discretized parameters obtained by using the classical finite element, the proposed method eliminates discretization procedures and thus, eliminates the process of forming mathematical models from additional errors associated with discretization procedures. This element of mathematical modeling of the dynamics of bending is called a harmonic element (HaE). The proposed method is based on the development of dynamic compliance methods, previously used to simulate oscillations of beam systems with distributed parameters. The developed mathematical models of stationary oscillations of thin elastic plates with distributed inertial parameters make it possible to include them in discrete-continuous models containing infinite-dimensional flexural elements (plates and beams), material points, solids and concentrated elasticities. Thus, dynamic models of structures subjected to harmonic effects are represented by a set of harmonic elements (HaE), which allow harmonic matching of heterogeneous elements under different boundary conditions.

Discretisation of regularity structures

Nous introduisons un cadre general permettant d’appliquer la theorie des structures de regularite a des discretisations d’EDP stochastiques. L’approche suivie dans cet article est que, au lieu de nous focaliser sur un type d’approximation specifique, nous supposons donnee une echelle $\varepsilon>0$ et une “boite noire” decrivant le comportement des objets discretises aux echelles plus petites.

An accelerated Boundary‐Domain Integral Method for three‐dimensional fluid flow analysis

AbstractThe rapid growth of computer power enables performing of more and more complex numerical simulations. However, the need to develop efficient numerical algorithms is always present. In this study, we propose an acceleration of an algorithm to solve the nonlinear system of Navier‐Stokes equations. We present an accelerated Boundary‐Domain Integral method (BDIM). The BDIM is a boundary element method (BEM) based method, thus its computational demands scale as 𝒪(N2), where N is the number of grid nodes. This is due to the fact that the discretization procedure leads to fully populated matrices. However, the computational demands can be reduced to a linear‐logarithmic 𝒪(N log N) complexity, with the implementation of an approximation procedure. In this study, we solve the velocity‐vorticity formulation of the Navier‐Stokes equations and develop an accelerated algorithm for determination of the boundary vorticity in 3D based on ℋ‐matrix and the adaptive cross approximation (ACA) method. We test the algorithm using coupled nanofluid flow and heat transfer. As expected results show that, the acceleration (matrix compression rate) has a negative influence on the solution accuracy.

A Maximum Entropy-Inspired Interpolative Closure for the Prediction of Radiative Heat Transfer in Laminar Co-Flow Diffusion Flames

ABSTRACT An interpolative-based first-order maximum entropy (M1) moment closure for providing approximate solutions to the equation of radiative transfer in non-gray participating media is proposed and described. This newly developed non-gray moment closure technique results in significant computational savings compared to an approach that makes use of the direct numerical solution of the optimization problem for entropy maximization. Its predictive capabilities are also assessed by comparing its solutions to those of the more commonly adopted first-order spherical harmonics (P1) moment closure technique, as well as the popular discrete ordinates method (DOM), which is used as a benchmark for the model comparisons. The evaluations are performed for sooting co-flow laminar diffusion flames for blends of ethanol and methane fuels at atmospheric as well as at elevated pressures and include comparisons to available experimental data for soot volume fraction and flame temperature. The strong spectral dependence exhibited by the absorption coefficient of real gases is treated herein using the statistical narrow-band correlated-k method, and the chemical kinetics of the underlying species are modeled using a reduced mechanism for methane and ethanol fuels. Theoretical details of the proposed interpolative M1, along with a description of the proposed Godunov-type finite-volume scheme developed for the numerical solution of the resulting system of hyperbolic moment equations are briefly discussed. The finite-volume method makes use of limited second-order solution reconstruction, multi-block body-fitted quadrilateral meshes with anisotropic adaptive mesh refinement (AMR), and an efficient Newton–Krylov–Schwarz (NKS) iterative method for solution of the resulting nonlinear algebraic equations arising from the finite-volume discretization procedure. The numerical results for laminar co-flow flames show that the flame solutions and predictions of soot formation of the non-gray M1 maximum-entropy moment closure are very promising and in very good agreement with those of both the DOM and P1 spherical harmonics model, while offering a substantial reduction in the number of dependent solution variables, compared to the DOM.

A stochastic network equilibrium model for electric power markets with uncertain demand

ABSTRACT We use the theory of stochastic variational inequalities to develop a network equilibrium model of the whole supply chain of electricity markets: power generation, supply, transmission, and consumption. In particular, we take into account the case where the market demand functions are not exactly known but are affected by some kind of uncertainty. A discretization and truncation procedure is used to numerically solve the stochastic variational inequality model. Monotonicity properties of the operator are investigated and the affine case is analysed in detail. Finally, numerical experiments show the impact of different probability densities of the random variables on the approximated solutions and the scalability of the proposed numerical method for real-world sized problems.

Scalable Graph-Based Clustering With Nonnegative Relaxation for Large Hyperspectral Image

Hyperspectral image (HSI) clustering is very important in remote sensing applications. However, most graph-based clustering models are not suitable for dealing with large HSI due to their computational bottlenecks: the construction of the similarity matrix $\boldsymbol {W}$ , the eigenvalue decomposition of the graph Laplacian matrix $\boldsymbol {L}$ , and $k$ -means or other discretization procedures. To solve this problem, we propose a novel approach, scalable graph-based clustering with nonnegative relaxation (SGCNR), to cluster the large HSI. The proposed SGCNR algorithm first constructs an anchor graph and then adds the nonnegative relaxation term. With this, the computational complexity can be reduced to $O(nd\log m+nK^{2}+nKc+K^{3})$ , compared with traditional graph-based clustering algorithms that need at least $O(n^{2}d+n^{2}K)$ or $O(n^{2}d+n^{3})$ , where $n$ , $d$ , $m$ , $K$ , and $c$ are, respectively, the number of samples, features, anchors, classes, and nearest neighbors. In addition, the SGCNR algorithm can directly obtain the clustering indicators, without resort to $k$ -means or other discretization procedures as traditional graph-based clustering algorithms have to do. Experimental results on several HSI data sets have demonstrated the efficiency and effectiveness of the proposed SGCNR algorithm.

Discrete analogues of continuous bivariate probability distributions

In many real-world applications, the phenomena of interest are continuous in nature and modeled through continuous probability distributions, but their observed values are actually discrete and hence it would be more reasonable and convenient to choose an appropriate (multivariate) discrete distribution generated from the underlying continuous model preserving one or more important features. In this paper, two methods are discussed for deriving a bivariate discrete probability distribution from a continuous one by retaining some specific features of the original stochastic model, namely (1) the joint density function, or (2) the joint survival function. Examples of applications are presented, which involve two types of bivariate exponential distributions, in order to illustrate how the discretization procedures work and show whether and to which extent they alter the dependence structure of the original model. We also prove that some bivariate discrete distributions that were recently proposed in the literature can be actually regarded as discrete counterparts of well-known continuous models. A numerical study is presented in order to illustrate how the procedures are practically implemented and to present inferential aspects. Two real datasets, considering correlated discrete recurrence times (the former) and counts (the latter) are eventually fitted using two discrete analogues of a bivariate exponential distribution.