Problems In Complex Geometries Research Articles

Adaptive multi-patch isogeometric analysis with truncated hierarchical B-splines in isotropic/orthotropic media

We present an adaptive multi-patch isogeometric analysis on the basis of truncated hierarchical B-splines (THB-splines) for solving two-dimensional complex isotropic/orthotropic elasticity. The THB-splines have local refinement property and their basis functions exhibit linear independence, so these features are highly applicable for adaptive isogeometric analysis. Guided by a posterior error estimator based on stress recovery, the adaptive algorithm is utilized in isogeometric analysis. In order to further extend the proposed method to solve complex geometry problems, the multi-patch technique is adopted to achieve exact modeling with Nitsche’s method as a multi-patch coupling approach. An isotropic numerical example with exact analytical solutions and three orthotropic numerical examples are presented to verify the effectiveness and accuracy of the developed method. Numerical solutions show that the developed adaptive isogeometric analysis method has high computational efficiency.

Low Mach Number Approximated Linearized Perturbed Euler Equations-based Unified Computational Flow-induced Acoustic Model for 2D Planar/Axisymmetric Complex Geometry

This article is on proposition of a unified 2D planar and axisymmetric differential formulation—for both fluid dynamics and acoustic governing equations. Further, using hydrodynamic splitting method for computational acoustics, a differential formulation is proposed by performing low Mach number approximation on Linearized Perturbed Euler Equations (LPEE). Finally, using the differential formulations, a unified numerical methodology is proposed for Computational Flow-induced Acoustics (CFiA)—involving complex geometry—on a body-fitted curvilinear grid. A finite volume and finite difference methods-based hybrid method is presented for algebraic formulation in CFiA. Further, solution methodology is presented with a semi-implicit pressure projection method for fluid flow, four step Runge-Kutta method along with sub time-stepping for acoustic perturbations, and a one-way explicitly coupled solution algorithm for the CFiA. A detailed validation study is presented separately for the present in-house computational-acoustic and CFiA solvers—by considering various test problems, involving 2D planar and axisymmetric complex geometry problems. Further, using an order of accuracy study, the present acoustic and CFiA solvers are demonstrated as IV-order and II-order accurate, respectively. Finally, using the CFiA solver, a superior performance is demonstrated for the proposed low Mach number approximated LPEE as compared to the traditional LPEE-based solver. For low Mach number CFiA, involving 2D planar and axisymmetric complex geometries, a novel method is presented here for computationally efficient CFiA development.

A general-purpose IGA mesh generation method: NURBS Surface-to-Volume Guided Mesh Generation

AbstractThe NURBS Surface-to-Volume Guided Mesh Generation (NSVGMG) is a general-purpose mesh generation method, introduced to increase the scope of isogeometric analysis in computing complex-geometry problems. In the NSVGMG, NURBS patch surface meshes serve as guides in generating the patch volume meshes. The interior control points are determined independent of each other, with only a small subset of the surface control points playing a role in determining each interior point. In the updated version of the NSVGMG we are introducing in this article, in the process of determining the location of an interior point in a parametric direction, more weight is given to the closer guides, with the closeness measured along the guides in the other parametric directions. Tests with 2D and 3D shapes show the effectiveness of the NSVGMG in generating good quality meshes, and the robustness of the updated NSVGMG even in mesh generation for complex shapes with distorted boundaries.

NG-DPSM: A neural green-distributed point source method for modelling ultrasonic field emission near fluid-solid interface using physics informed neural network

Distributed point source method (DPSM) is a collocation point-based semi-analytical method to model the scattered ultrasonic fields in isotropic and anisotropic materials. It requires the evaluation of Green's function solutions and its rigorous differentiation for complex geometry problems having a large set of distributed point sources. DPSM is much faster than the finite element method (FEM) however, it is analytically demanding as it requires differentiation of the Green's function. The intricacy associated with differentiation operation hinders the potential application of DPSM in large-scale automation and real-time structural health monitoring. The current paper introduces machine-learning-based strategies within DPSM and yields the proposed Neural Green-DPSM (NG-DPSM) framework. A physics-informed neural network (PINN) is incorporated in the DPSM framework for evaluating the Green's function and its gradients. To train the PINN, displacement Green's function solutions of wave propagation in isotropic solid is used as the governing physics. Once the PINN is trained for displacement Green's function solutions, the NG-DPSM framework leverages the trained network and its automatic differentiation capability to predict displacement and stress fields annihilating the rigorous differentiation of Green's functions. The accuracy and efficacy of NG-DPSM are demonstrated using two numerical experiments. First, the ultrasonic fields are evaluated for the problem geometry of a plate immersed in fluid excited at different angles of incidence. Further, the efficacy of the proposed approach is demonstrated for wave scattering through a circular hole in the plate. The results show a strong agreement between the ultrasonic fields computed using both NG-DPSM and conventional DPSM.

The smooth extension embedding method with Chebyshev polynomials

AbstractWe propose an implementation of the smooth extension embedding method (SEEM), first described by Agress and Guidotti's study, in the setting of Chebyshev polynomials. SEEM is a hybrid fictitious domain/collocation method which solves general boundary value problems in complex domains by recasting them as constrained optimization problems in a simple encompassing set. Previously, SEEM was introduced and implemented using a periodic box (read a torus) using Fourier series; here, it is implemented on a (non‐periodic) rectangle using Chebyshev polynomial expansions. This implementation has faster convergence on smaller grids. Numerical experiments will demonstrate that the method provides a simple, robust, efficient, and high order fictitious domain method which can solve elliptic and parabolic problems in complex geometries, with non‐constant coefficients, and for general boundary conditions. We consider applications to two and three dimensional boundary value problems as well as an initial boundary value problem via a genuinely space–time discretization.

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.

A computational approach based on extended finite element method for thin porous layers in acoustic problems

AbstractThe simulation of stationary acoustic fields involving thin porous layers is addressed in the present article. Layer thickness is assumed to be relatively thin in comparison to the overall computational domain, but its non‐negligible acoustic impact must be taken into consideration in the numerical model. Within the classical finite element method (FEM), meshes are compatible at material interfaces and the element distortions needs to be avoided. These requirements usually lead to an excessively costly spatial discretization for these problems of interest, as it forces mesh refinement surrounding the thin layer. This article provides a computational approach to relax this restriction. A generalized interface model derived from the plane wave transfer matrix Method (TMM) is established for modeling thin layers. We develop variationally consistent formulations to impose the interface conditions from models of thin layers for diverse coupling configurations, in which both acoustic fluid and poro‐elastic media are taken into account. The computational domain is discretized using the extended finite element method (X‐FEM) in order to introduce strong discontinuities in elements independently of the mesh. Implementation of the proposed formulations within X‐FEM is verified to be capable of providing accurate and robust solutions. The efficiency and flexibility of the present approach for multi‐layer and complex geometry problems are demonstrated compared to classical interface‐fitted finite element models through different simulation scenarios.

Domain decomposition involving subdomain separable space representations for solving parametric problems in complex geometries

A domain decomposition technique combined with an enhanced geometry mapping based on the use of NURBS is considered for solving parametrized models in complex geometries (non simply connected) within the so-called proper generalized decomposition (PGD) framework, enabling the expression of the solution in each subdomain in a fully separated form, involving both the space and the model parameters. NURBS allow a compact and powerful domain mapping into a fully separated reference geometry, while the PGD allows recovering an affine structure of the problem in the reference domain, facilitating the use of the standard PGD solver for computing the parametric solution in each subdomain first, and then by enforcing the interface transmission conditions, in the whole domain.

The third golden age of aeroacoustics

The present review covers the latest evolution of computational aeroacoustics, the field that deals with the noise generated by fluid flows and its propagation in the medium. It highlights the latest findings in both free flows (jet noise) and wall-bounded flows (airfoil, airframe, and turbomachinery noise) in more and more complex environments. Among the computational aero-acoustics methods, high-order schemes of the Navier–Stokes equations on unstructured grids and the lattice Boltzmann method on Cartesian grids have emerged as excellent candidates to tackle noise problems in realistic complex geometries. The latter is also shown to be particularly efficient for both noise generation and propagation, allowing to directly estimate the noise in the far field. Two examples of application of such methods to complex jet noise and to installed airfoil noise are first presented. The first one involves compressible subsonic and supersonic flows in dual-stream nozzles and the second one subsonic flow around an airfoil embedded in the potential core of the open-jet anechoic wind tunnel as in the actual trailing-edge noise experiment. For airframe noise, large eddy simulations of scaled nose landing gear noise and three-element high-lift devices can be tackled to decipher noise sources. For turbomachinery noise, simulations of installed low-speed fans have already unveiled a wealth of details on their noise sources, whereas high-speed turbofans remain a challenge giving the high Reynolds numbers and small tip gaps involved.

Volume Penalization Method for Solving Coupled Radiative-Conductive Heat Transfer Problems in Complex Geometries

Volume Penalization Method for Solving Coupled Radiative-Conductive Heat Transfer Problems in Complex Geometries

2D finite elements for the computational analysis of crack propagation in brittle materials and the handling of double discontinuities

Crack growth simulations by way of the traditional Finite Element Method claim progressive remeshing to fit the geometry of the fracture, severely increasing the computational effort. Methods such as the eXtended Finite Element Method (XFEM) allow to overcome this limitation by means of nodal shape functions multiplied by Heaviside step function to enrich finite element nodes. Through the medium of a discontinuous field, the entire geometry of the discontinuity can be modelled regardless of the mesh, avoiding remeshing. In this paper two shell-type XFEM elements (a three-node triangular element and a four-node quadrangular element) to evaluate crack propagation in brittle materials are presented. These elements have been implemented into the widespread opensource framework OpenSees to evaluate crack propagation into a plane shell subjected to monotonically increasing loads. Moreover, in the perspective of fracture propagation simulations, the problem of managing multiple cracks without remeshing or operating subdivisions on the integration domain has been investigated and a four-node quadrangular finite element for the computational analysis of double crossed discontinuities by the means of equivalent polynomials is presented in this paper. Equivalent polynomials allow to overcome inaccuracies on the results when performing standard numerical integration (e.g. Gauss-Legendre quadrature rule) over the entire domain of XFEM elements, without the need of defining integration subdomains. The presented work and the computational strategy behind it may be extremely useful not only in the field of fracture mechanics, but also to solve complex geometry problems or material discontinuities.

Study on the Formation Mechanism of Surface Adhered Damage in Ball-End Milling Ti6Al4V.

Ball-end cutters are widely used for machining the parts of Ti-6Al-4V, which have the problem of poor machined surface quality due to the low cutting speed near the tool tip. In this paper, through the experiments of inclined surface machining in different feed directions, it is found that the surface adhered damages will form on the machined surface under certain tool postures. It is determined that the formation of surface adhered damage is related to the material adhesion near the cutting edge and the cutting-into/out position within the tool per-rotation cycle. In order to analyze the cutting-into/out process more clearly under different tool postures, the projection models of the cutting edge and the cutter workpiece engagement on the contact plane are established; thus, the complex geometry problem of space is transformed into that of plane. Combined with the case of cutting-into/out, chip morphology, and surface morphology, the formation mechanism of surface adhered damage is analyzed. The analysis results show that the adhered damage can increase the height parameters Sku, Sz, Sp, and Sv of surface topographies. Sz, Sp, and Sv of the normal machined surface without damage (Sku ≈ 3) are about 4–6, 2–3, and 2–3 μm, while Sz, Sp, and Sv with adhered damage (Sku > 3) can reach about 8–20, 4–14, and 3–6 μm in down-milling and 10–25, 7–18, and 3–7 μm in up-milling. The feed direction should be selected along the upper left (Q2: β ∈ [0°, 90°]) or lower left (Q3: β ∈ [90°, 180°]) to avoid surface adhered damage in the down-milling process. For up-milling, the feed direction should be selected along the upper right (Q1: β ∈ (−90°, 0°]) or upper left (Q2: β ∈ [0°, 90°)).

A fixed grid based accurate phase-field method for dendritic solidification in complex geometries

In this paper, an easily implementable immersed interface method based high-resolution phase-field model is proposed for simulating dendritic growth related problems in complicated geometries. At first, resolution issues of different Fourier Pseudo-spectral based dealiasing schemes are shown for the anisotropic phase field equations at high thermal undercooling conditions using just adequate spatial grid resolution. The problems of aliasing error and spectral leakage are studied using high-order based exponential smoothing spectral filters, which generally occur at low sampling rate situations. A fully dealiased zero padding scheme based anisotropic phase-field method is also developed in conjunction with an optimal third order strong stability preserving Runge–Kutta (SSPRK3) time integration scheme for obtaining more accurate and stable numerical solutions. In the later part, a modified phase-field method is implemented with the indicator function which simulates different crystal growth problems in complex geometries without using non-trivial mesh refinement algorithm. Effect of non-interacting obstacles present in the computational domain can also be readily incorporated in the present numerical model by using the single order parameter based phase-field equations.

SOLVING TRANSIENT INVERSE HEAT TRANSFER PROBLEMS IN COMPLEX GEOMETRIES USING PHYSICS-GUIDED NEURAL NETWORKS (PGNN)

Temporally and spatially unstable thermal conditions lead to transient or inhomogeneous thermo-elastic behavior of workpieces during manufacturing or geometric inspection. Temperature monitoring by means of sensors consign transient temperature fields, but do not yield information about the heat flow acting as thermal boundary condition, which is a relevant input parameter for nearly any thermal simulation. Addressing the need for efficient methods, the authors propose an approach to solve inverse heat transfer problems in complex geometries. In the presented study, locally acting heat loads are experimentally investigated based on virtual demonstrators running in FEM. The conducted method shows high potential for transient heat flow modelling in terms of accuracy and computational efficiency.

Discrete unified gas kinetic scheme simulation of conjugate heat transfer problems in complex geometries by a ghost-cell interface method

In this paper, we extend the discrete unified gas kinetic scheme (DUGKS) to simulate conjugate heat transfer flows. The conjugate interface that separates two media with different thermophysical properties is handled by a ghost-cell (GC) immersed boundary method. Specifically, fictitious cells are set in both domains to implement the continuity conditions of both temperature and heat flux at the conjugate interface. Information at the ghost cells is unavailable and should be reconstructed from the interface and neighboring field. With interface constraints involved in the reconstruction, the conjugate condition can be incorporated smoothly into the process of solving the full flow field. Those features make the present GC-DUGKS capture the conjugate interface, without introducing an additional term or modifying the equilibrium distribution function as in previous studies. Furthermore, the Strang-splitting scheme is applied for convenient handling of the thermal force term in the governing equation. Simulations of several well-established natural convection flows containing complex conjugate interfaces are performed to validate the GC-DUGKS. The results demonstrate the accuracy and feasibility of the present method for conjugate heat transfer problems.

Application of the immersed boundary method in solution of radiative heat transfer problems

The immersed boundary method (IBM) was developed for the first time in the present study to simulate the radiative heat transfer problems in complex geometries. The pseudo time stepping technique was applied to solve the radiative transfer equation (RTE) using the finite volume method (FVM). Also, the direct forcing-sharp interface was used in IBM. This method was validated with some benchmark problems in two states of pure radiative and combined radiative-conductive heat transfer. Then, the effects of the conduction-radiation parameter and absorption coefficient on heat flux distribution and isotherms were investigated in an enclosure with inner complex bodies. The results indicated that this method acts well in solving the radiative heat transfer problems in complex geometries. Also, due to using unique Eulerian and Lagrangian grids for solving the energy and radiative transfer equations, production of a separate grid is not required in solving combined heat transfer problems. This is one of the most significant features of the presented method.

Mixed Kirchhoff stress–displacement–pressure formulations for incompressible hyperelasticity

The numerical approximation of hyperelasticity must address nonlinear constitutive laws, geometric nonlinearities associated with large strains and deformations, the imposition of the incompressibility of the solid, and the solution of large linear systems arising from the discretisation of 3D problems in complex geometries. We adapt the three-field formulation for nearly incompressible hyperelasticity introduced in Chavan et al. (2007) to the fully incompressible case. The mixed formulation is of Hu–Washizu type and it differs from other approaches in that we use the Kirchhoff stress, displacement, and pressure as principal unknowns. We also discuss the solvability of the linearised problem restricted to neo-Hookean materials, illustrating the interplay between the coupling blocks. We construct a family of mixed finite element schemes (with different polynomial degrees) for simplicial meshes and verify its error decay through computational tests. We also propose a new augmented Lagrangian preconditioner that improves convergence properties of iterative solvers. The numerical performance of the family of mixed methods is assessed with benchmark solutions, and the applicability of the formulation is further tested in a model of cardiac biomechanics using orthotropic strain energy densities. The proposed methods are advantageous in terms of physical fidelity (as the Kirchhoff stress can be approximated with arbitrary accuracy and no locking is observed) and convergence (the discretisation and the preconditioners are robust and computationally efficient, and they compare favourably at least with respect to classical displacement–pressure schemes).

A discrete collocation scheme to solve Fredholm integral equations of the second kind in high dimensions using radial kernels

In this paper, we propose a kernel-based method to solve multidimensional linear Fredholm integral equations of the second kind over general domains. The discrete collocation method in combination with radial kernels interpolation method is utilized to convert these types of equations to a linear system of equations that can be solved numerically by a suitable numerical method. Integrals appeared in the scheme are approximately computed by the Gauss–Legendre and Monte Carlo quadrature rules. The proposed scheme does not require a structured grid, and thus can be used to solve complex geometry problems based on a set of scattered points that can be arbitrarily chosen. Thus, for the multidimensional linear Fredholm integral equation, an irregular region can be considered. The convergence analysis of the approach is studied for the presented method. The accuracy and efficiency of the new technique are illustrated by several numerical examples.

Plasma-induced catalysis: towards a numerical approach.

A lattice Boltzmann (LB) model is developed, validated and used to study simplified plasma/flow problems in complex geometries. This approach solves a combined set of equations, namely the Navier-Stokes equations for the momentum field, the advection-diffusion and the Nernst-Planck equations for electrokinetic and the Poisson equation for the electric field. This model allows us to study the dynamical interaction of the fluid/plasma density, velocity, concentration and electric field. In this work, we discuss several test cases for our numerical model and use it to study a simplified plasma fluid flowing and reacting inside a packed bed reactor. Inside the packed bed, electric breakdown reactions take place due to the electric field, making neutral species ionize. The presence of the packed beads can help enhance the reaction efficiency by locally increasing the electric field, and the size of packed beads and the pressure drop of the packed bed do influence the outflux. Hence trade-offs exist between reaction efficiency and packing porosity, the size of packing beads and the pressure drop of the packed bed. Our model may be used as a guidance to achieve higher reaction efficiencies by optimizing the relevant parameters. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

An accurate triangular spectral element method-based numerical simulation for acoustic problems in complex geometries

An accurate triangular spectral element method (TSEM) is developed to simulate acoustic problems in complex computational domains. With Fekete points and Koornwinder-Dubiner polynomials introduced, triangular elements are used in the present method to substitute quadrilateral elements in traditional spectral element method (SEM). The efficiency of discretizing complex geometry is enhanced while high accuracy of SEM is remained. The weak form of the second-order governing equations derived from the linearized Euler equations (LEEs) are solved, and perfectly matched layer (PML) boundary condition is implemented. Three benchmark problems with analytical solutions are employed to testify the exponential convergence rate, convenient implementation of solid wall boundary condition and capable discretization in complex geometries of the present method respectively. An application on Helmholtz resonator (HR) is presented as well to demonstrate the possibility of using the present method in practical engineering. The numerical resonance frequency of HR reaches an excellent agreement with the theoretical result.