Dense Discretization Research Articles

Mesh-Splitting Impedance Transition Boundary Condition for Accurate Modeling of Thin Structures

Numerical modeling of thin structures can result in dense volumetric discretization and poor mesh quality, which can be computationally expensive. A few equivalent boundary conditions are able to replace thin structures with zero-thickness surfaces and avoid volumetric discretization. However, their near-field solutions suffer from loss of accuracy as the thickness increases, because the reduction of thin structures to zero-thickness surfaces changes the original problem. This work presents a mesh-splitting technique for the impedance transition boundary condition (ITBC), which effectively eliminates the aforementioned error without additional number of unknowns. Under certain assumptions, the technique can be easily applied and yield much more accurate solutions than several other equivalent boundary conditions; otherwise, it may be unfeasible or lead to marginal improvements. Moreover, the mesh-splitting ITBC is incorporated into the spectral-element spectral-integral method for the modeling of doubly periodic problems including metasurfaces. Numerical examples include error analysis for the proposed technique and demonstrate its potential in more complex structures.

Efficient physics‐based recurrent neural network model for radio wave propagation in tunnels at 2.4 GHz

AbstractThe vector parabolic equation (VPE) method has been widely applied to the modelling of radio wave propagation in tunnel environments. However, dense spatial discretizations are generally required for VPE simulations to achieve acceptable accuracy. This results in high computational costs for large‐size tunnels and high‐frequency wireless systems. This letter presents an efficient physics‐based recurrent neural network (RNN) model for modeling radio wave propagation in tunnels. The proposed RNN‐based model can leverage coarse‐mesh VPE to generate high‐fidelity received signal strength and achieve significant computational savings. The validity of the proposed approach is demonstrated against fine‐mesh VPE simulations in various tunnel cases.

Modeling of the Achilles Subtendons and Their Interactions in a Framework of the Absolute Nodal Coordinate Formulation.

Experimental results have revealed the sophisticated Achilles tendon (AT) structure, including its material properties and complex geometry. The latter incorporates a twisted design and composite construction consisting of three subtendons. Each of them has a nonstandard cross-section. All these factors make the AT deformation analysis computationally demanding. Generally, 3D finite solid elements are used to develop models for AT because they can discretize almost any shape, providing reliable results. However, they also require dense discretization in all three dimensions, leading to a high computational cost. One way to reduce degrees of freedom is the utilization of finite beam elements, requiring only line discretization over the length of subtendons. However, using the material models known from continuum mechanics is challenging because these elements do not usually have 3D elasticity in their descriptions. Furthermore, the contact is defined at the beam axis instead of using a more general surface-to-surface formulation. This work studies the continuum beam elements based on the absolute nodal coordinate formulation (ANCF) for AT modeling. ANCF beam elements require discretization only in one direction, making the model less computationally expensive. Recent work demonstrates that these elements can describe various cross-sections and materials models, thus allowing the approximation of AT complexity. In this study, the tendon model is reproduced by the ANCF continuum beam elements using the isotropic incompressible model to present material features.

Iso-Geometric Integral Equation Solvers and Their Compression via Manifold Harmonics

The state of art of electromagnetic integral equations has seen significant growth over the past few decades, overcoming some of the fundamental bottlenecks: computational complexity, low frequency and dense discretization breakdown, preconditioning, and so on. Likewise, the community has seen extensive investment in development of methods for higher order analysis, in both geometry and physics. Unfortunately, these standard geometric descriptors are continuous, but their normals are discontinuous at the boundary between triangular tessellations of control nodes, or patches, with a few exceptions; as a result, one needs to define additional mathematical infrastructure to define physical basis sets for vector problems. In stark contrast, the geometric representation used for design are second order differentiable almost everywhere on the surfaces. Using these description for analysis opens the door to several possibilities, and is the area we explore in this paper. Our focus is on Loop subdivision based isogeometric methods. In this paper, our goals are two fold: (i) development of computational infrastructure for isogeometric analysis of electrically large simply connected objects, and (ii) to introduce the notion of manifold harmonics transforms and its utility in computational electromagnetics. Several results highlighting the efficacy of these two methods are presented.

Identifying Moving Vibrational Sources in a Truncated, Damped, Heterogeneous Solid

This paper discusses a novel, robust, computational framework for reconstructing spatial and temporal profiles of moving vibrational sources in a heterogeneous, elastic, damped, truncated one-dimensional solid using sparsely measured wave responses. We use the finite element method to obtain wave solutions because of its flexibility and robustness for heterogeneous media. To reconstruct wave source profiles without a priori knowledge of the sources, we employ high-resolution discretization of source functions in space and time. Because of such dense discretization, the order of magnitude of the number of inversion parameters could range up to hundreds of thousands. To identify such a large number of control parameters, an adjoint-gradient-based source inversion approach is used within a context of discretization-then-optimization (DTO). Numerical experiments prove the robustness of this method by reconstructing spatial and temporal profiles of multiple dynamic moving body forces in a heterogeneous, damped solid bar. The numerical experiments show that using the conjugate gradient method gives improved results over the steepest descent method. The inversion performance is not affected by the acceleration, frequency, or amplitude of targeted moving dynamic distributed loads. While inversion performance is not affected by the damping or wave speed in the domain when the model is homogeneous, a mismatch in acoustic impedance for materials in a heterogeneous solid bar leads the inversion to converge more slowly. The inversion is sensitive to noise, but filtering the noise from the measured data help reduce the inversion error.

A Large Scale Benchmark for the Incompressible Navier-Stokes Equations

We introduce a collection of benchmark problems in 2D and 3D (geometry description and boundary conditions), including simple cases with known analytic solution, classical experimental setups, and complex geometries with fabricated solutions for evaluation of numerical schemes for incompressible Navier-Stokes equations in laminar flow regime. We compare the performance of a representative selection of most broadly used algorithms for Navier-Stokes equations on this set of problems. Where applicable, we compare the most common spatial discretization choices (unstructured triangle/tetrahedral meshes and structured or semi-structured quadrilateral/hexahedral meshes). The study shows that while the type of spatial discretization used has a minor impact on the accuracy of the solutions, the choice of time integration method, spatial discretization order, and the choice of solving the coupled equations or reducing them to simpler subproblems have very different properties. Methods that are directly solving the original equations tend to be more accurate than splitting approaches for the same number of degrees of freedom, but numerical or computational difficulty arise when they are scaled to larger problem sizes. Low-order splitting methods are less accurate, but scale more easily to large problems, while higher-order splitting methods are accurate but require dense time discretizations to be stable. We release the description of the experiments and an implementation of our benchmark, which we believe will enable statistically significant comparisons with the state of the art as new approaches for solving the incompressible Navier-Stokes equations are introduced.

An efficient FEM-BEM solution for electromagnetic analysis of inhomogeneous objects

An efficient finite element method-boundary element method (FEM-BEM) is proposed for electromagnetic analysis of inhomogeneous objects. The problem domain is discretized and evaluated with FEM, and the BEM introduces a truncated boundary on the surface of the domain utilizing boundary integral equations (BIEs) technique. To guarantee the precision, much dense discretization is usually required when modeling and simulating the inhomogeneous media, which will lead to a large and dense impedance matrix from the BEM part. Therefore, the nested complex source beam (NCSB) method is proposed in the FEM-BEM system to accelerate the evaluation of the BIEs, where the interaction between far-field groups is equivalent to that of the complex source beams located on the equivalent sphere of each group in an aggregation, translation and disaggregation process. Furthermore, the adaptive grouping technique for the NCSB is employed to improve the performance of the solution. Numerical results show the efficiency of the proposed method.

A Multi-Variate framework to assess reliability and discrimination power of Bayesian estimation of Intravoxel Incoherent Motion parameters.

To propose a multivariate multi-step framework for a systematic assessment of the estimation reliability and discriminability of Intravoxel Incoherent Motion (IVIM) model parameters. Monte-Carlo simulations were generated on a range of SNRs and in different IVIM combinations considering: i) a dense discretization with 24 b-values; ii) a discretization with 9 b-values. A state-of-the-art Bayesian fitting method was adopted. The framework assessed: i) the best model between mono- and bi-exponential, through the BIC index; ii) the fitting accuracy; iii) the power in discriminating two different IVIM parameters distributions of estimated coefficients, using a multivariate test. Exemplificative oncologic cases were also presented. The bi-exponential fitting was reliable for perfusion fraction higher than 5%, with high accuracy in D estimation, acceptable error for f, but high uncertainty in D*. The discrimination of two distributions is generally feasible if differences in D values (at least 0.3 x10-3 mm2/s) are present; in the case of similar D values, a minimal difference of 5% in f can be discriminated just in case of balanced sample size and dense b-values discretization, whereas the impact of D* is quite negligible. These results were also supported by clinical examples. IVIM model is generally accurate in estimating diffusion, but uncertainties related to perfusion estimation are not negligible and compromise the discrimination power when different populations should be differentiated. The proposed framework should be adopted as interpretative guidelines to better understand when IVIM model applied on real data can provide reliable findings.

Synthesis of control Lyapunov functions and stabilizing feedback strategies using exit‐time optimal control Part I: Theory

AbstractThis work studies the problem of constructing control Lyapunov functions (CLFs) and feedback stabilization strategies for deterministic nonlinear control systems described by ordinary differential equations. Many numerical methods for solving the Hamilton–Jacobi–Bellman partial differential equations specifying CLFs typically require dense state space discretizations and consequently suffer from the curse of dimensionality. A relevant direction of attenuating the curse of dimensionality concerns reducing the computation of the values of CLFs and associated feedbacks at any selected states to finite‐dimensional nonlinear programming problems. We propose to use exit‐time optimal control for that purpose. This article is the first part of a two‐part work. First, we state an exit‐time optimal control problem with respect to a sublevel set of an appropriate local CLF and establish that, under a number of reasonable conditions, the concatenation of the corresponding value function and the local CLF is a global CLF in the whole domain of asymptotic null‐controllability. We also investigate the formulated optimal control problem. A modification of these constructions for the case when one does not find a suitable local CLF is provided as well. Our developments serve as a theoretical basis for a curse‐of‐dimensionality‐free approach to feedback stabilization, that is presented in the second part Yegorov et al. (2021) of this work together with supporting numerical simulation results.

Complete greenhouse dynamic simulation tool to assess the crop thermal well-being and energy needs

Greenhouse technology is an important energy consumer sector representing an indispensable solution for modern methods of crop production. In the greenhouse envelope and system design phase, thermodynamic performance simulation tools are required. The greenhouse simulation is still a very complex task despite many building tools are available in the literature.This work aims to overcome this gap by proposing a reference methodology to accommodate the building TRNSYS software for a greenhouse able to consider simultaneously different thermal phenomena with detailed modelling of: dense volume discretization, 3D shortwave and longwave radiative exchange, air flow exchanges, presence of lamps with their exact 3D position, ground and plant evapotranspiration, and convective heat transfer coefficients.A standard hourly simulation of the one-zone greenhouse was validated with another recognized tool, showing excellent agreement throughout the year. The main parameters affecting the greenhouse thermal balance were investigated in both a free-floating and continuous regime. The investigation has shown that a standard simulation is accurate to only reproduce the thermal response in a free-floating regime; instead, the detailed simulation has led to overall cooling and heating energy needs in the continuous regime, respectively, of 51.4 kWh/m3 and 49.1 kWh/m3, avoiding to obtain very high errors.

Discretization of the Bloch sphere, fractal invariant sets and Bell's theorem.

An arbitrarily dense discretization of the Bloch sphere of complex Hilbert states is constructed, where points correspond to bit strings of fixed finite length. Number-theoretic properties of trigonometric functions (not part of the quantum-theoretic canon) are used to show that this constructive discretized representation incorporates many of the defining characteristics of quantum systems: completementarity, uncertainty relationships and (with a simple Cartesian product of discretized spheres) entanglement. Unlike Meyer’s earlier discretization of the Bloch Sphere, there are no orthonormal triples, hence the Kocken–Specker theorem is not nullified. A physical interpretation of points on the discretized Bloch sphere is given in terms of ensembles of trajectories on a dynamically invariant fractal set in state space, where states of physical reality correspond to points on the invariant set. This deterministic construction provides a new way to understand the violation of the Bell inequality without violating statistical independence or factorization, where these conditions are defined solely from states on the invariant set. In this finite representation, there is an upper limit to the number of qubits that can be entangled, a property with potential experimental consequences.

An order N log N parallel solver for time-spectral problems

The time-spectral method is a fast and efficient scheme for computing the solution to temporal periodic problems. Compared to traditional backward difference implicit time-stepping methods, time-spectral methods incur significant computational savings by using a temporal Fourier representation of the time discretization and solving the periodic problem directly. In the time-spectral discretization, all time instances are fully coupled to each other, resulting in a dense temporal discretization matrix, the evaluation of which scales as O(N2), where N denotes the number of time instances. However, by implementing the time-spectral method based on the fast Fourier transform (FFT) the computational cost can be reduced to O(Nlog⁡N). Furthermore, in parallel implementations, where each time instance is assigned to an individual processor, the wall-clock time necessary to solve time-spectral problems is reduced to O(log⁡N) using the FFT-based approach, as opposed to the O(N) weak scaling incurred by previous dense matrix or discrete Fourier transform (DFT) parallel time-spectral solver implementations. In this work, first an FFT-based approximate factorization (AF) scheme is used to solve time-spectral problems with large numbers of time instances. Subsequently, this solution strategy is reformulated as a preconditioner to be used in the context of a Newton-Krylov method applied directly to the complete non-linear space-time time-spectral residual. The use of the Generalized Minimal Residual Method (GMRES) Krylov method enables additional coupling between the various time instances running on different processors resulting in faster overall convergence. The GMRES/AF scheme is shown to produce significantly faster convergence than the AF scheme used as a solver alone, and achieves orders of magnitude gains in efficiency compared to previous DFT-based implementations of similar solution strategies for large numbers of time instances.

A multi-discretization scheme for topology optimization based on the parameterized level set method

In the framework of the parameterized level set method, the structural analysis and topology representation can be implemented in a decoupling way. A parameterized level set function, typically, using radial basis functions (RBFs), is a linear combination of a set of prescribed RBFs and coefficients. Once the coefficients are determined, the theoretical level set function is determined. Exploiting this inherent property, we propose a multi-discretization method based on the parameterized level set method. In this approach, a coarse discretization is applied to do the structural analysis whereas another dense discretization is employed to represent the structure topology. As a result, both efficient analysis and high-resolution topological design are available. Note that the dense discretization only accounts for a more precise and smooth description of the theoretical level set function rather than introduce extra design freedom or incur interference to structural analysis or the optimization process. In other words, this decoupling way will not add to the computational burden of structural analysis or result in non-uniqueness of converged results for a particular analysis setting. Numerical examples in both two-dimension and three-dimension show effectiveness and applicability of the proposed method.

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.

Fast solution of elliptic harbor agitation problems under frequency-direction input spectra by model order reduction and NURBS-enhanced FEM

Many harbor applications are based on the solution of linear elliptic agitation problems for many spectral conditions. One of the main goals consists in computing the linear combination of numerous simulations of the harbor agitation problem, using monochromatic waves of different spectral components (i.e. frequency and incoming wave direction). In practice, the standard strategy selects the number of wave components according to a prescribed discretization of the 2D input spectra. The main issue relies on some quantities of interest that are very sensitive to the level of refinement of the spectra, such as the significant wave height at every mesh point or the identification of resonance modes induced by long wave scattering. In many cases, achieving enough quality in these quantities may impose numerous simulations and, consequently, non-practical computer costs. This can drastically limit the final accuracy of results. To overcome this situation, here a new strategy is proposed to efficiently solve a large number of harbor agitation problems derived from dense discretizations of the 2D input spectra. The strategy is based on the combination of two different numerical approaches. Firstly, each required monochromatic simulation is solved via high order NURBS (non-uniform rational B-splines) enhanced finite elements (NEFEM). More precisely, NEFEM captures the exact harbor geometry using large mesh elements that produce accurate solutions and significant savings on the system size, particularly in long wave cases. Secondly, a model order reduction technique is used to approximate the original elliptic harbor model by a so-called surrogate model. The main advantage is that, once the surrogate model is constructed, it can be rapidly evaluated to provide simulations for any value of the spectral components within a range of interest, and without the need of solving any new harbor agitation problem (as the standard strategy does). Thus, this enables the possibility of using any desired discretization of the 2D input spectra with no additional computer cost. The construction of the surrogate model is performed using the proper generalized decomposition method with a novel incremental computation along the frequency dimension. The proposed strategy is discussed, and its superior performance with respect to standard strategies is demonstrated, on two harbor agitation examples with several applications.

Implementation of the Equivalence Principle Algorithm for Potential Integral Equations

A novel implementation of the equivalence principle algorithm (EPA) employing potential integral equations (PIEs) is presented. EPA is generalized to be compatible with PIEs that are used to formulate inner problems inside equivalence surfaces. Based on the stability of PIEs, the resulting EPA-PIE implementation is suitable for low-frequency problems involving dense discretizations with respect to wavelength. Along with the formulation and a clear demonstration of the EPA-PIE mechanism, high accuracy, stability, and efficiency of the implementation are presented on canonical problems.

Parallel Wideband MLFMA for Analysis of Electrically Large, Nonuniform, Multiscale Structures

Electromagnetic scattering from electrically large objects with multiscale features is an increasingly important problem in computational electromagnetics. A conventional approach is to use an integral equation-based solver that is then augmented with an accelerator, a popular choice being a parallel multilevel fast multipole algorithm (MLFMA). One consequence of multiscale features is locally dense discretization, which leads to low-frequency breakdown and requires nonuniform trees. To the authors' knowledge, the literature on parallel MLFMA for such multiscale distributions capable of arbitrary accuracy is sparse; this paper aims to fill this niche. We prescribe an algorithm that overcomes this bottleneck. We demonstrate the accuracy (with respect to analytical data) and performance of the algorithm for both PEC scatterers and point clouds as large as 755λ with several hundred million unknowns and nonuniform trees as deep as 16 levels.

When integration sparsification fails: Banded Galerkin discretizations for Hermite functions, rational Chebyshev functions and sinh-mapped Fourier functions on an infinite domain, and Chebyshev methods for solutions with C[formula omitted] endpoint singularities

Chebyshev polynomial spectral methods are very accurate, but are plagued by the cost and ill-conditioning of dense discretization matrices. Modified schemes, collectively known as “integration sparsification”, have mollified these problems by discretizing the highest derivative as a diagonal matrix. Here, we examine five case studies where the highest derivative diagonalization fails. Nevertheless, we show that Galerkin discretizations do yield banded matrices that retain most of the advantages of “integration sparsification”. Symbolic computer algebra greatly extends the reach of spectral methods. When spectral methods are implemented using exact rational arithmetic, as is possible for small truncation N in Maple, Mathematica and their ilk, roundoff error is irrelevant, and sparsification failure is not worrisome. When the discretization contains a parameter L, symbolic algebra spectral methods return, as answer to an eigenproblem, not discrete numbers but rather a plane algebraic curve defined as the zero set of a bivariate polynomial P(λ,L); the optimal approximations to the eigenvalues λj are in the middle of the straight portions of the zero contours of P(λ;L) where the isolines are parallel to the L axis.

Numerical simulation of the cavitating KP505 propeller working in open water conditions

The present paper proposes a numerical study, fully experimentally validated, usable for the optimal design of the naval propulsion features. The marine propeller KP 505 is a representative propeller geometry with experimental available detailed, so this is chosen as a reference for probing the accuracy of the present study. Data obtained from this model is extensively used to provide validation for unsteady RANS calculations, with specific interest in predicting tip-vortex flow characteristics. The method used is based on the fundamental conservation equations. Since DNS modeling requires excessive computational effort due to the very dense discretization of computational domain and very limited time step, it is assumed that the influence of turbulence on the flow is derived with sufficient accuracy based on the averaged flow parameters. In the present computations the EASM model is applied, which proved to be a stable and accurate model. The computed overall accuracy of the proposed model is validated by an extensive comparison with experimental data from the free running tests performed by NMRI Tokyo and HSVA Hamburg. The main features of the flow field, with particular attention to the vortical structures detached from the blades are comparatively presented in an attempt to validate the technique proposed.

Fast Diffeomorphic Image Registration via Fourier-Approximated Lie Algebras

This paper introduces Fourier-approximated Lie algebras for shooting (FLASH), a fast geodesic shooting algorithm for diffeomorphic image registration. We approximate the infinite-dimensional Lie algebra of smooth vector fields, i.e., the tangent space at the identity of the diffeomorphism group, with a low-dimensional, bandlimited space. We show that most of the computations for geodesic shooting can be carried out entirely in this low-dimensional space. Our algorithm results in dramatic savings in time and memory over traditional large-deformation diffeomorphic metric mapping algorithms, which require dense spatial discretizations of vector fields. To validate the effectiveness of FLASH, we run pairwise image registration on both 2D synthetic data and real 3D brain images and compare with the state-of-the-art geodesic shooting methods. Experimental results show that our algorithm dramatically reduces the computational cost and memory footprint of diffemorphic image registration with little or no loss of accuracy.