Problems In Complex Geometries Research Articles

Using DC PSE operator discretization in Eulerian meshless collocation methods improves their robustness in complex geometries

Many fluid-dynamics applications require solutions in complex geometries. In these cases, mesh generation can be a difficult and computationally expensive task for mesh-based methods. This is alleviated in meshless methods by relaxing the neighborhood relations between nodes. Meshless methods, however, often face issues computing numerically robust local operators, especially for the irregular node configurations required to effectively resolve complex geometries. Here we address this issue by using Discretization-Corrected Particle Strength Exchange (DC PSE) operator discretization in a strong-form Eulerian collocation meshless solver. We use the solver to compute steady-state solutions of incompressible, laminar flow problems in standard benchmarks and multiple complex-geometry problems in 2D with a velocity-correction method in the Eulerian framework. We verify that the solver produces stable and accurate results across all benchmark problems. We find that DC PSE operator discretization is more robust to varying node configurations than Moving Least Squares (MLS). In addition, we find that in more challenging complex geometries, the solver using MLS operator discretization fails to converge, whereas DC PSE operators provide robust solutions without node adjustment.

Modelling ground vibrations induced by harmonic loads

A finite-element model combining the frequency domain thin-layer method with paraxial boundary conditions to simulate the semi-infinite extent of a soil medium is presented in this paper. The combined numerical model is used to deal with harmonic vibrations of surface rigid foundations on non-horizontal soil profiles. The model can deal with soil media over rigid bedrock or significant depths of half-space. Structured finite elements are used to mesh simple geometry soil domains, whereas unstructured triangular mesh grids are employed to deal with complex geometry problems. Dynamic responses of homogeneous as well as layered soil profiles are simulated and validated against analytical and approximate solutions. Finally, the model is used to deal with surface ground vibration reduction, in which it is first validated against published results and then followed by an example involving a bridge.

Tropical Aspects in Geometry, Topology and Physics

The workshop Tropical Aspects in Geometry, Topology and Physics was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject. The development of tropical geometry is based on deep links between problems in real and complex enumerative geometry, symplectic geometry, quantum fields theory, mirror symmetry, dynamical systems and other research areas. On the other hand, new interesting phenomena discovered in the framework of tropical geometry (like refined tropical enumerative invariants) pose the problem of a conceptual understanding of these phenomena in the “classical” geometry and mathematical physics.

A meshless singular boundary method for three‐dimensional elasticity problems

SummaryThis study documents the first attempt to extend the singular boundary method, a novel meshless boundary collocation method, for the solution of 3D elasticity problems. The singular boundary method involves a coupling between the regularized BEM and the method of fundamental solutions. The main idea here is to fully inherit the dimensionality and stability advantages of the former and the meshless and integration‐free attributes of the later. This makes it particularly attractive for problems in complex geometries and three dimensions. Four benchmark 3D problems in linear elasticity are well studied to demonstrate the feasibility and accuracy of the proposed method. The advantages, disadvantages, and potential applications of the proposed method, as compared with the FEM, BEM, and method of fundamental solutions, are also examined and discussed. Copyright © 2015 John Wiley & Sons, Ltd.

Octree-based Multiple-material Parallel Unstructured Mesh Generation Method for Seismic Response Analysis of Soil-Structure Systems

We developed an unstructured finite element mesh generation method capable of modeling multiple-material complex geometry problems for large-scale seismic analysis of soil-structure systems. We used an octree structure to decompose the target domain into small subdomains and use the multiple material marching cubes method for robust and parallel tetrahedralization of each subdomain. By using the developed method on a 32 core shared memory machine, we could generate a 594,168,792 tetrahedral element soil-structure model of a power plant in 13h 01min. The validity of the generated model was confirmed by conducting a seismic response analysis on 2,304 compute nodes of the K computer at RIKEN. Although the model contains a small approximation in geometry (half of the minimum octree size) at present, we can expect fast and high quality meshing of large-scale models by making geometry correction in the future, which is expected to help improve the seismic safety of important structures and complex urban systems.

Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries

The aim of this work is to solve parametrized partial differential equations in computational domains represented by networks of repetitive geometries by combining reduced basis and domain decomposition techniques. The main idea behind this approach is to compute once, locally and for few reference shapes, some representative finite element solutions for different values of the parameters and with a set of different suitable boundary conditions on the boundaries: these functions will represent the basis of a reduced space where the global solution is sought for. The continuity of the latter is assured by a classical domain decomposition approach. Test results on Poisson problem show the flexibility of the proposed method in which accuracy and computational time may be tuned by varying the number of reduced basis functions employed, or the set of boundary conditions used for defining locally the basis functions. The proposed approach simplifies the pre-computation of the reduced basis space by splitting the global problem into smaller local subproblems. Thanks to this feature, it allows dealing with arbitrarily complex network and features more flexibility than a classical global reduced basis approximation where the topology of the geometry is fixed.

Simulation of steady fluid–solid conjugate heat transfer problems via immersed boundary-lattice Boltzmann method

In this paper, we extend the lattice Boltzmann method (LBM) in combination with the immersed boundary method (IBM) to simulate the steady-state fluid–solid conjugate heat transfer problems in complex geometries. The no-slip boundary condition is handled by the immersed boundary method. The “pseudo-solid-specific-heat” technique is used to ensure the continuity of heat flux at the fluid–solid interface. The present LB thermal field solver retains the simple structure of standards thermal LB model. The performance of the present method is validated by several heat conduction/convection problems. The numerical results agree well with exact solutions or published results.

Meshless modeling of natural convection problems in non-rectangular cavity using the variational multiscale element free Galerkin method

In this paper, the two-dimensional natural convection problems in complex geometries were solved by using the variational multiscale element free Galerkin (VMEFG) method. The VMEFG method is a meshless method which coupled element free Galerkin method and variational multiscale method, thus it inherits the advantages of variational multiscale and meshless methods. In this method, the field variables are decomposed into coarse and fine scales first, then solved fine scale problem analytically by using bubble functions, in the process, the stabilization parameters had appeared naturally. Moreover, it ensures that the resultant formulations yield a consistent stabilized method. From the viewpoint of application, the presented method can employ equal order basis for pressure and velocity, which is not only easy to implement but also avoid the restriction of the Babuŝka–Brezzi condition and eliminate non-physical oscillations. Several test problems, including natural convection in the semicircular cavity, triangular cavity and triangular cavity with zig-zag shaped bottom wall are considered to investigate the accuracy of the proposed method. The numerical results obtained using VMEFG showed very good agreement with those available in the literature.

Fast-multipole accelerated singular boundary method for large-scale three-dimensional potential problems

The singular boundary method (SBM) is a relatively new meshless boundary collocation method for the numerical solution of certain elliptic boundary value problems. The method involves a coupling between the boundary element method (BEM) and the method of fundamental solutions (MFS). The main idea is to fully inherit the dimensionality and stability advantages of the former and the meshless and integration-free attributes of the later. This makes it particularly attractive for problems in complex geometries and three dimensions. However, similar to the traditional BEM, the SBM produces dense and unsymmetrical coefficient matrices, which requires O(N2) memory and another O(N3) operations to solve the system with direct solvers. This paper documents the first attempt to apply the fast multipole method (FMM) to accelerate the solutions of the SBM for the solution of large-scale problems. The FMM formulations for the SBM are presented for three-dimensional (3D) potential problems. Numerical examples with up to 900,000 unknowns are solved successfully on a desktop computer using the developed FMM-SBM code. These results clearly demonstrate the efficiency, accuracy and potentials of the FMM-SBM for solving large-scale problems.

Swath-acquisition planning in multiple-satellite missions: an exact and heuristic approach

This paper deals with the swath acquisition planning problem for multisatellite Earth observation missions. Given a set of satellites and a mission time frame, the problem we solve consists of selecting a set of acquisitions from the satellites in order to cover a given region of interest during the requested time frame, optimizing a certain objective function.We show that the planning problem can be modeled as a set covering problem, using basic tools of mathematical programming. The formulation of the model requires the solution of a complex computational geometry problem, and therefore the use of heuristics and metaheuristics applies. In this paper, we discuss the efficiency of the constructive phase of a greedy randomized adaptive search procedure algorithm. Computational results comparing the heuristic algorithms with the exact approach are presented.

A 2D software system for expedite analysis of CFD problems in complex geometries

ABSTRACTEasyCFD_G, a 2D software tool for fluid dynamics and heat transfer applications, is presented. The underlying physical and numerical principles are described, including mesh generation. The software is based on quadrilateral structured and unstructured meshes, rendering it with great versatility in geometrical terms. Amenable problems include multi‐domain simulations with fluid and solid regions, thermal effects and mixing of different fluids, in stationary or turbulent regime. For illustration, a few example problems are presented, covering different physical situations, including comparison with benchmark data. Due to its versatility, fast learning curve and easy and intuitive utilization, this software tool represents a great value for educational purposes. © 2015 Wiley Periodicals, Inc. Comput. Appl. Eng. Educ. 24:27–38, 2016; View this article online at wileyonlinelibrary.com/journal/cae; DOI 10.1002/cae.21668

Methods of solving complex geometry problems

1 Problems Involving Triangles.- 2 Quadrilaterals and other Polygons.- 3 Problems Involving Circles.- 4 Problems on Construction.- Appendix A Ratios and Proportion.- Appendix B My 9th Grade Notebook Page.- Appendix C My Pictures.- References.- Index.

DEM Extraction in Urban Areas Using High-Resolution TerraSAR-X Imagery

Three-dimensional (3D) spatial information is crucial for improving the quality of human life through urban planning and management, and it is widely utilized due to its rapid, periodic and inexpensive acquisition. In this context, extraction of digital surface and elevation models (DSM and DEM) is a significant research topic for space-borne optical and synthetic aperture radar (SAR) remote sensing. The DSMs include visible features on the earth’s surface such as vegetation, forest and elevated man-made objects, while DEMs contain only the bare ground. In this paper, using TerraSAR-X (TSX) high resolution Spotlight (HS) images, high-resolution interferometric DEM generation in a part of Istanbul urban area is aimed. This is not an easy task because of SAR imaging problems in complex geometry of urban settlements. The interferometric processing steps for DSM generation were discussed including critical parameters and thresholds to improve the quality of the final product and a 3 m gridded DSM was generated. The DSM-DEM conversion was performed by filtering and the quality of generated DEM was verified against a reference DEM from stereo photogrammetry with 3 m original grid spacing. The achieved root mean square error of height differences (RMSZ) varies from 7.09 to 8.11 m, depending on the terrain slope. The differential DEM, illustrates the height differences between generated DEM and the reference DEM, was generated to show the correlation between height differences and the coherence map. Finally, a perspective view of test area was created draping extracted DEM and a high-resolution IKONOS panchromatic image.

Numerical Simulation of Polyurethane Foaming Process Using Finite Point Method

For the numerical simulation of fluid mechanics problems in complex geometries, the use of the classical grid methods such as the finite element method and the finite volume method can give rise to several problems related to the deformation of the mesh. In this work, a meshfree Lagrangian method is used to avoid these problems. This method called Finite Point Method (FPM) has been developed by Kuhnert. It consists in representing the fluid domain by a set of particles. The efficiency of this method is pointed by studying the problem of polyurethane foaming. To do so, we have adopted a theoretical model describing the contribution of the chemical kinetics and the rheological coupling characterizing such process. This coupling is displayed through the dependency of the fluid viscosity to the evolution of the temperature and the chemical reactions present during the foaming process. The expansion of the mixture is governed by the front velocity which is calculated by solving the Navier-Stokes equations. Compared to the experimental results for polyurethane foaming process in a conical beaker, the numerical results using the FPM code are acceptable.

The MFS for the solution of harmonic boundary value problems with non-harmonic boundary conditions

We investigate applications of the method of fundamental solutions (MFS) for the numerical solution of two-dimensional boundary value problems in complex geometries, governed by the Laplace equation and subject to Dirichlet boundary conditions which are not harmonic. Such problems can be very challenging because of the appearance of boundary singularities. We consider several ways of choosing the boundary collocation points as well as the source points in the MFS. We show that with an appropriate such choice the MFS yields highly accurate results.

Adaptive Runge–Kutta discontinuous Galerkin method for complex geometry problems on Cartesian grid

SUMMARYA Cartesian grid method using immersed boundary technique to simulate the impact of body in fluid has become an important research topic in computational fluid dynamics because of its simplification, automation of grid generation, and accuracy of results. In the frame of Cartesian grid, one often uses finite volume method with second order accuracy or finite difference method. In this paper, an h‐adaptive Runge–Kutta discontinuous Galerkin (RKDG) method on Cartesian grid with ghost cell immersed boundary method for arbitrarily complex geometries is developed. A ghost cell immersed boundary treatment with the modification of normal velocity is presented. The method is validated versus well documented test problems involving both steady and unsteady compressible flows through complex bodies over a wide range of Mach numbers. The numerical results show that the present boundary treatment to some extent reduces the error of entropy and demonstrate the efficiency, robustness, and versatility of the proposed approach. Copyright © 2013 John Wiley & Sons, Ltd.

Accurate optical model for design and analysis of solar fields based on heterogeneous multicore systems

This work deals with parallel algorithms for the optical design and analysis of solar fields for Central Receiver Systems. The proposed implementation based on desktop workstations equipped with multiple Graphics Processing Units (GPU) is motivated by the need to have an accurate and fast simulation environment for studying mirror imperfection and non-planar geometries. Since heliostat field layouts can be composed of thousands of mirrors, and for each one a complex 3D geometry problem must be solved, this can take a long time even with highly optimized CPU-based solvers. The GPU-accelerated solver outperforms an optimized OpenMP-based reference, running on 8 CPU cores, up to 52× depending on the complexity of the problem, at the cost of a very modest hardware upgrade of two GPU GTX 590 and one GTX 480 graphics cards. Several applications demonstrate that the parallel algorithm preserves flexibility to model complex mirror geometries and non-idealities.

High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

We present a review of some of the state-of-the-art numerical methods for solving the Stefan problem and the Poisson and the diffusion equations on irregular domains using (i) the level-set method for representing the (possibly moving) irregular domain's boundary, (ii) the ghost-fluid method for imposing the Dirichlet boundary condition at the irregular domain's boundary and (iii) a quadtree/octree node-based adaptive mesh refinement for capturing small length scales while significantly reducing the memory and CPU footprint. In addition, we highlight common misconceptions and describe how to properly implement these methods. Numerical experiments illustrate quantitative and qualitative results.

Prediction of radiative heat transfer in 2D irregular geometries using the collocation spectral method based on body-fitted coordinates

Recently, an efficient numerical method, which is called the collocation spectral method (CSM), for radiative heat transfer problems, has been proposed by the present authors. In this numerical method there exists the exponential convergence rate, which can obtain a very high accuracy even using a small number of grids. In this article, the CSM based on body-fitted coordinates (BFC) is extended to simulate radiative heat transfer problems in participating medium confined in 2D complex geometries. This numerical method makes simultaneously the use of the merits of both the CSM and BFC. In this numerical approach, the radiative transfer equation (RTE) in orthogonal Cartesian coordinates should be transformed into the equation in body-fitted nonorthogonal curvilinear coordinates. In order to test the efficiency of the developed method, several 2D complex irregular enclosures with curved boundaries and containing an absorbing, emitting and scattering medium are examined. The results obtained by the CSM are assessed by comparing the predictions with those in references. These comparisons indicate that the CSM based on BFC can be recommended as a good option to solve radiative heat transfer problems in complex geometries.

Upwind-Difference Potentials Method for Patlak-Keller-Segel Chemotaxis Model

We develop a novel upwind-difference potentials method for the Patlak-Keller-Segel chemotaxis model that can be used to approximate problems in complex geometries. The chemotaxis model under consideration is described by a system of two nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. Chemotaxis is an important process in many medical and biological applications, including bacteria/cell aggregation and pattern formation mechanisms, as well as tumor growth. Furthermore modeling of real biomedical problems often has to deal with the complex structure of computational domains. There is consequently a need for accurate, fast, and computationally efficient numerical methods for different chemotaxis models that can handle arbitrary geometries. The upwind-difference potentials method proposed here handles complex domains with the use of only Cartesian meshes, and can be easily combined with fast Poisson solvers. In the numerical tests presented below we demonstrate the robustness of the proposed scheme.