Hierarchical Basis Research Articles

三维薄结构热传导问题分层二次元方程的多水平方法

When the finite element method is applied to analyze the 3D thin-walled structures, some thin hexahedral elements are usually used in order to reduce the number of elements, and the corresponding higher-order elements are preferred since they have some obvious advantages in the calculation accuracy, the anti-distortion degree and so on. However, they have much higher computational complexity than the lower-order (e.g., linear) elements and the coefficient matrix of the linear algebraic equation system is severely ill-conditioned. The convergence of the commonly used solvers will deteriorate with the increasing size of the problem. An efficient and robust multi-level method was presented for the hierarchical quadratic discretizations of 3D thin-walled structures through combination of two special local block Gauss-Seidel smoothers and the DAMG algorithm based on the distance matrix. Since a hierarchical basis is used, those algebraic criteria are not needed to judge the relationships between the unknown variables and the geometric node types, and the grid transfer operators are also trivial. This makes it easy to find the coarse level (linear element) matrix derived directly from the fine level matrix, and thus the overall efficiency is greatly improved. The numerical results verify the efficiency and robustness of the proposed method.

<b>Urban Form that Minimizes the Total Travel Cost Assuming Multiple Floors in a Three-Dimensional City with Two-Stage Hierarchical Bases</b>

<b>Urban Form that Minimizes the Total Travel Cost Assuming Multiple Floors in a Three-Dimensional City with Two-Stage Hierarchical Bases</b>

T-Ω Formulation with Higher-Order Hierarchical Basis Functions for Nonsimply Connected Conductors

This paper presents in detail the extension of the T-Ω formulation for eddy currents based on higher-order hierarchical basis functions so that it can automatically deal with conductors of arbitrary topology. To this aim, we supplement the classical hierarchical basis functions with nonlocal basis functions spanning the first de Rham cohomology group of the insulating region. Such nonlocal basis functions may be efficiently and automatically found in negligible time with the recently introduced Dłotko–Specogna (DS) algorithm. The approach presented in this paper merges techniques together which to some extent already existed in literature but they were never grouped together and tested as a single unit.

Improved progressive morphological filter for digital terrain model generation from airborne lidar data.

Obtaining high-precision filtering results from airborne lidar point clouds in complex environments has always been a hot topic. Mathematical morphology was widely used for filtering, owing to its simplicity and high efficiency. However, the morphology-based algorithms are deficient in preserving terrain details. In order to obtain a better filtering effect, this paper proposed an improved progressive morphological filter based on hierarchical radial basis function interpolation (PMHR) to refine the classical progressive morphological filter. PMHR involved two main improvements, namely, automatic setting of self-adaptive thresholds and terrain details preservation, respectively. The performance of PMHR was evaluated using datasets provided by the International Society for Photogrammetry and Remote Sensing. Experimental results show that PMHR achieved good performance under variant terrain features with an average total error of 4.27% and average Kappa coefficient of 84.57%.

Personalized Care and the Role of Insulin as a Vehicle to Optimizing Treatments in Diabetes Care.

In patients with type 2 diabetes (T2D) with poor glycemic control, there is an unmet need for treatment optimization involving the initiation and/or intensification of insulin therapy, which is often delayed because of clinical inertia. Educational initiatives that target patients and physicians might be one way to address this need. To evaluate the effectiveness of educational materials mailed to physicians and their patients in affecting initiation of insulin therapy and other health care outcomes. This study, named PIVOTs (Personalized care and the role of Insulin as a Vehicle to Optimizing Treatments), used integrated medical and pharmacy claims data from the U.S.-based HealthCore Integrated Research Database between January 1, 2006, and April 4, 2014, to identify patients who were potential candidates for insulin therapy. Eligible patients were aged 18-75 years, currently enrolled in a commercial or Medicare Advantage health plan, with T2D diagnosis codes. Patients selected for insulin treatment education had glycated hemoglobin A1c (A1c) > 10%, irrespective of the number of noninsulin antihyperglycemic drugs used, or A1c > 8.0% and ≤ 10% while receiving ≥ 2 noninsulin antihyperglycemic drugs. For each identified patient, a corresponding treating physician was identified on a hierarchical basis. Physician-level randomization was conducted to assign physicians and their linked patients to the following 4 cohorts: (1) a cross-sectional cohort in which educational materials were sent to patients and physicians on a single outreach date; (2) a longitudinal cohort in which educational materials were sent to patients and physicians on 2 occasions, 3 months apart; (3) an enhanced cohort in which patients and physicians received the same mailings as the longitudinal cohort, plus physicians were invited to attend a 1:1 video conference academic detailing session; and (4) a control cohort in which patients and physicians did not receive any educational materials. Insulin initiation rates, A1c levels, and medical and pharmacy costs were assessed from claims over 6 and 12 months follow-up within each cohort. Mean insulin initiation rates at 12 months ranged from 9.2%-10.3% (all patients) to 12.3%-14.9% (subset with baseline A1c ≥ 9.0%), with similar rates across the intervention and control cohorts. Reductions in A1c from baseline were also similar across cohorts for all patients (0.1%-0.6%), as well as for those with a baseline A1c ≥ 9.0% (0.9%-1.6%). Approximately 14%-20% of patients achieved A1c < 7.0%, with no differences across cohorts. Changes in mean total all-cause and diabetes-related health care costs were also similar across cohorts. The findings of this real-world, randomized intervention call into question the value of educational mailings as a means to overcoming clinical inertia and improving health outcomes in patients with T2D, at least in the context of insulin initiation. This study was funded by Sanofi US. Bieszk and Wei are employees of Sanofi US. Grabner, Barron, and Quimbo are employees of HealthCore, which was under contract with Sanofi US for the conduct of this study. Yan is an employee of PHAR, LLC and was employed by HealthCore at the time this study was conducted and completed. Biel is an employee of Anthem. Chu is a consultant for Sanofi US; a member of the lecture bureaus for AstraZeneca, Eli Lilly, and Sanofi US; and has received research funding from Novo Nordisk. Study concept and design were contributed by Bieszk, Grabner, Wei, Quimbo, and Barron. Yan, Barron, Quimbo, and Grabner collected the data, which were interpreted by Biel, Chu, Bieszk, and Wei, with assistance from the other authors. The manuscript was written by Bieszk, with assistance from the other authors, and revised by Bieszk, Grabner, Biel, and Chu, along with the other authors. Part of this work was presented in poster format at the 76th Scientific Sessions of the American Diabetes Association; June 10-14, 2016; New Orleans, Louisiana.

Superconvergence analysis of second and third order rectangular edge elements with applications to Maxwell’s equations

Superconvergence for the second and third order edge elements is investigated on nonuniform rectangular meshes. First, we develop the explicit expression for the Nédélec interpolation based on the hierarchical basis functions. Then we prove that the pointwise interpolation error estimates are one order higher at element Gauss points than the standard analysis can provide. Using the superconvergence at Gauss points, we establish the discrete l2 norm superconvergence for the solution of Maxwell’s equations solved by both the second and third order rectangular edge elements. Numerical results justifying our theoretical analysis are presented.

An efficient high order plane wave time domain algorithm for transient electromagnetic scattering analysis

An efficient high order plane wave time domain algorithm is presented for analyzing the transient scattering from three dimensional electrically large conducting objects. This method uses a set of hierarchical divergence-conforming vector basis functions to accurately represent the current distribution on the perfect electrically conducting (PEC) surface. The higher order functions can significantly reduce the number of unknowns without compromise on the accuracy. The time domain combined field integral equation (TD-CFIE) is then discretized using the hierarchical divergence-conforming vector basis functions and shifted Lagrange polynomial functions in spatial and time domain, respectively. The final matrix equation can be accelerated using the plane wave time domain (PWTD) algorithm. Finally, a parallel algorithm that can execute on a distributed-memory parallel cluster is developed, which provides an appealing avenue for analyzing the transient scattering from three-dimensional electrically large complex PEC objects. Numerical examples are given to demonstrate the accuracy and efficiency of the method.

Recovery-based error estimation and adaptivity using high-order splines over hierarchical T-meshes

An adaptive higher-order method based on a generalization of polynomial/rational splines over hierarchical T-meshes (PHT/RHT-splines) is introduced. While most problems considered in isogeometric analysis can be solved efficiently when the solution is smooth, many non-trivial simulations have rough solutions. This can be caused, for example, by the presence of re-entrant corners in the domain. For such problems, a tensor-product basis is less suitable for resolving the singularities that appear, as refinement propagates throughout the computational domain. Hierarchical bases and adaptivity allow for a more efficient way of dealing with singularities, by adding more degrees of freedom only where they are necessary to improve the approximation. In order to drive the adaptive refinement, an efficient recovery-based error estimator is proposed in this work. The estimator produces a “recovered solution” which is a more accurate approximation than the computed numerical solution. Several 2D and 3D numerical investigations with PHT-splines of higher order and greater continuity show good performance compared to uniform refinement in terms of degrees of freedom and computational cost.

2-D High-Order DGTD Method and Its Application in Analysis of Sheath Propagation Characteristics

A 2-D discontinuous Galerkin time-domain (DGTD) scheme based on high-order hierarchical basis function is presented in this paper. Combined with the shift operator method solving the electromagnetic problem of dispersive media in finite-difference time domain, the iterative formula of DGTD method in dealing with plasma media is given. On this basis, taking the example of line element radiation, the computing ability of different basis functions is compared. To study the transmission characteristics of the electromagnetic (EM) wave in sheath, the DGTD calculation models of the plasma sheath of a blunt cone at different flight heights and velocities are established using Lagrange interpolation technique based on the calculation results of COMSOL (business software). The influences of antenna position, flight altitude, and flight speed on transmission characteristics of the EM wave are discussed through calculation and analyses.

Fast analysis of large-scale problems by modified multilevel compressed block decomposition combined with high order hierarchical basis functions

Fast analysis of large-scale problems by modified multilevel compressed block decomposition combined with high order hierarchical basis functions

Reducing the cost of using collocation to compute vibrational energy levels: Results for CH2NH.

In this paper, we improve the collocation method for computing vibrational spectra that was presented in the work of Avila and Carrington, Jr. [J. Chem. Phys. 143, 214108 (2015)]. Known quadrature and collocation methods using a Smolyak grid require storing intermediate vectors with more elements than points on the Smolyak grid. This is due to the fact that grid labels are constrained among themselves and basis labels are constrained among themselves. We show that by using the so-called hierarchical basis functions, one can significantly reduce the memory required. In this paper, the intermediate vectors have only as many elements as the Smolyak grid. The ideas are tested by computing energy levels of CH2NH.

Nonconformal Discretization of Electric Current Volume Integral Equation With Higher Order Hierarchical Vector Basis Functions

A nonconformal discretization of the electric current volume integral equation (JVIE) is presented for the electromagnetic scattering analysis of inhomogeneous dielectric objects. This JVIE is based on higher order geometrical and current modeling, where curved tetrahedral elements and higher order hierarchical vector (HOHV) basis functions are adopted. In order to implement the nonconformal discretization, the half face-based HOHV basis function is introduced to address the nonconformal volumetric elements. Compared to the conventional low-order conformal/nonconformal VIE, our nonconformal HOHV-based JVIE has the advantage of improving the computational efficiency and accuracy. Compared to the previous VIE discretized by other higher order functions, our nonconformal JVIE is based on the higher order geometric modeling and the HOHV bases. In order to improve the efficiency, the basis expansion and recombination technique is introduced to significantly accelerate the matrix filling. Moreover, the flexibility of basis order selection is further enhanced by the mixed order schemes. Numerical results are given to demonstrate its accuracy, efficiency, and flexibility.

A Third Order Hierarchical Basis WENO Interpolation for Sparse Grids with Application to Conservation Laws with Uncertain Data

In this paper, we introduce a third order hierarchical basis WENO interpolation, which possesses similar accuracy and stability properties as usual WENO interpolations. The main motivation for the hierarchical approach is the direct applicability on sparse grids. This is for instance of large practical interest in the numerical solution of conservation laws with uncertain data, where discontinuities in the physical domain often carry over to the (potentially high-dimensional) stochastic domain. For this, we apply the introduced hierarchical basis WENO interpolation within a non-intrusive collocation method and present first results on 2- and 3-dimensional sparse grids.

A goal-based angular adaptivity method for thermal radiation modelling in non grey media

This paper investigates for the first time a goal-based angular adaptivity method for thermal radiation transport, suitable for non grey media when the radiation field is coupled with an unsteady flow field through an energy balance. Anisotropic angular adaptivity is achieved by using a Haar wavelet finite element expansion that forms a hierarchical angular basis with compact support and does not require any angular interpolation in space. The novelty of this work lies in (1) the definition of a target functional to compute the goal-based error measure equal to the radiative source term of the energy balance, which is the quantity of interest in the context of coupled flow-radiation calculations; (2) the use of different optimal angular resolutions for each absorption coefficient class, built from a global model of the radiative properties of the medium. The accuracy and efficiency of the goal-based angular adaptivity method is assessed in a coupled flow-radiation problem relevant for air pollution modelling in street canyons. Compared to a uniform Haar wavelet expansion, the adapted resolution uses 5 times fewer angular basis functions and is 6.5 times quicker, given the same accuracy in the radiative source term.

Energy consistent framework for continuously evolving 3D crack propagation

This paper presents an enhanced theoretical formulation and associated computational framework for brittle fracture in elastic solids within the context of configurational mechanics, building on the authors’ previous paper, Kaczmarczyk et al. (2014). The local form of the first law of thermodynamics provides an equilibrium condition for the crack front, expressed in terms of the configurational forces. Applying the principle of maximal energy dissipation, it is shown that the direction of the crack propagation is given by the direction of the configurational forces. In combination with a fracture criterion, these are utilised to determine the position of the continuously evolving crack front. This exploitation of the crack front equilibrium condition leads to a completely new, implicit, crack propagation formulation. A monolithic solution strategy is adopted, solving simultaneously for both the material displacements (i.e. crack extension) and the spatial displacements. The resulting crack path is resolved as a discrete displacement discontinuity, where the material displacements of the nodes on the crack front change continuously, without the need for mesh splitting or the use of enrichment techniques. In order to trace the dissipative loading path, an arc-length procedure is adopted that controls the incremental crack area growth. In order to maintain mesh quality, smoothing of the mesh is undertaken as a continuous process, together with face flipping, node merging and edge splitting where necessary. Hierarchical basis functions of arbitrary polynomial order are adopted to increase the order of approximation without the need to change the finite element mesh. Performance of the formulation is demonstrated by means of three representative numerical simulations, demonstrating both accuracy and robustness.

High Order Hierarchical Divergence-Free Constrained Transport H(div) Finite Element Method for Magnetic Induction Equation

AbstractIn this paper, we propose to use the interior functions of an hierarchical basis for high order BDMp elements to enforce the divergence-free condition of a magnetic field B approximated by the H(div)BDMp basis. The resulting constrained finite element method can be used to solve magnetic induction equation in MHD equations. The proposed procedure is based on the fact that the scalar (p–1)-th order polynomial space on each element can be decomposed as an orthogonal sum of the subspace defined by the divergence of the interior functions of the p-th order BDMp basis and the constant function. Therefore, the interior functions can be used to remove element-wise all higher order terms except the constant in the divergence error of the finite element solution of the B-field. The constant terms from each element can be then easily corrected using a first order H(div) basis globally. Numerical results for a 3-D magnetic induction equation show the effectiveness of the proposed method in enforcing divergence-free condition of the magnetic field.

A new high-order finite volume method for 3D elastic wave simulation on unstructured meshes

In this paper, we proposed a new efficient high-order finite volume method for 3D elastic wave simulation on unstructured tetrahedral meshes. With the relative coarse tetrahedral meshes, we make subdivision in each tetrahedron to generate a stencil for the high-order polynomial reconstruction. The subdivision algorithm guarantees the number of subelements is greater than the degrees of freedom of a complete polynomial. We perform the reconstruction on this stencil by using cell-averaged quantities based on the hierarchical orthonormal basis functions. Unlike the traditional high-order finite volume method, our new method has a very local property like DG and can be written as an inner-split computational scheme which is beneficial to reducing computational amount. Moreover, the stencil in our method is easy to generate for all tetrahedrons especially in the three-dimensional case. The resulting reconstruction matrix is invertible and remains unchanged for all tetrahedrons and thus it can be pre-computed and stored before time evolution. These special advantages facilitate the parallelization and high-order computations. We show convergence results obtained with the proposed method up to fifth order accuracy in space. The high-order accuracy in time is obtained by the Runge–Kutta method. Comparisons between numerical and analytic solutions show the proposed method can provide accurate wavefield information. Numerical simulation for a realistic model with complex topography demonstrates the effectiveness and potential applications of our method. Though the method is proposed based on the 3D elastic wave equation, it can be extended to other linear hyperbolic system.

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

We introduce a family of implementations of low-order, additive, geometric multilevel solvers for systems of Helmholtz equations arising from Schrödinger equations. Both grid spacing and arithmetics may comprise complex numbers, and we thus can apply complex scaling to the indefinite Helmholtz operator. Our implementations are based on the notion of a spacetree and work exclusively with a finite number of precomputed local element matrices. They are globally matrix-free. Combining various relaxation factors with two grid transfer operators allows us to switch from additive multigrid over a hierarchical basis method into a Bramble-Pasciak-Xu (BPX)-type solver, with several multiscale smoothing variants within one code base. Pipelining allows us to realize full approximation storage (FAS) within the additive environment where, amortized, each grid vertex carrying degrees of freedom is read/written only once per iteration. The codes realize a single-touch policy. Among the features facilitated by matrix-free FAS is arbitrary dynamic mesh refinement (AMR) for all solver variants. AMR as an enabler for full multigrid (FMG) cycling—the grid unfolds throughout the computation—allows us to reduce the cost per unknown. The present work primary contributes toward software realization and design questions. Our experiments show that the consolidation of single-touch FAS, dynamic AMR, and vectorization-friendly, complex scaled, matrix-free FMG cycles delivers a mature implementation blueprint for solvers of Helmholtz equations in general. For this blueprint, we put particular emphasis on a strict implementation formalism as well as some implementation correctness proofs.

A High-Order Time Domain Discontinuous Galerkin Method with Orthogonal Tetrahedral Basis for Electromagnetic Simulations in 3-D Heterogeneous Conductive Media

AbstractWe present a high-order discontinuous Galerkin (DG) method for the time domain Maxwell's equations in three-dimensional heterogeneous media. New hierarchical orthonormal basis functions on unstructured tetrahedral meshes are used for spatial discretization while Runge-Kutta methods for time discretization. A uniaxial perfectly matched layer (UPML) is employed to terminate the computational domain. Exponential convergence with respect to the order of the basis functions is observed and large parallel speedup is obtained for a plane-wave scattering model. The rapid decay of the out-going wave in the UPML is shown in a dipole radiation simulation. Moreover, the low frequency electromagnetic fields excited by a horizontal electric dipole (HED) and a vertical magnetic dipole (VMD) over a layered conductive half-space and a high frequency ground penetrating radar (GPR) detection for an underground structure are investigated, showing the high accuracy and broadband simulation capability of the proposed method.

Numerical Modeling for Gully Erosion in the North of Algeria based on Data of Remote Sensing and Geographic Information Systems

The problems of soil erosion are largely widespread in the countries of the Mediterranean basin. The process of gullying is a complex phenomenon with disastrous consequences. It particularly affects northern Algeria, decreasing the potentialities of the water tanks, reducing cultivable lands availability and degrading infrastructures. Therefore, this work studies the analysis and the prediction of gullying erosion by using a probabilistic approach based on multisource data. The objective of this search is to answer to the three following questions: i) which factors support the process of gullying ? ii) how does a process of gullying develop? iii) which are the zones favourable to gullying ? Works are undertaken on the catchment area of the Isser River. We focused the applications on the upstream part of the basin. In this research, we study a North-South transect which corresponds to three under-basins slopes. The choice of these tests areas answers to four criteria defined in our method: the representativeness, the homogeneity, the availability of former data and, finally, the accessibility. After the completion of the multisource data, modelling and multivariate analysis for the prediction of gullying. The combination factor-process by the univariate analysis allows on the one hand, to highlight the variables controlling the process of gullying, and on the other hand, to analyse the variables on a hierarchical basis and to know their degree of influence. The multivariate analysis, by the logistic regression model (LRM), enabled us to select the significant variables and to locate the most favourable zones for the process of gullying. The validation of the models is evaluated using the curves of lift spin. The results suggest that the factors highlighted by the model to be most influential on gullying erosion are: the lithology, the slope, the morphopedology, the rainfall erosivity and the land cover. The synthesis of this approach is illustrated in the form of charts of gullying erosion risk maps in four classes of probability. The assessment of the study shows the fundamental interest of this approach using geographical information systems and remote sensing, in particular for the watersheds of the southern Mediterranean, with the possibility of extending this methodology to other regions.