Hierarchical Basis Research Articles

A unified framework for the multi-scale computational homogenisation of 3D-textile composites

This paper extends the applications of a novel and fully automated multi-scale computational homogenisation framework, originally proposed by the authors (Ullah et al. (2017)) for unidirectional and 2D-textile composites, to 3D-textile composites. 3D-textile composites offer many advantages over 2D-textile composites but their highly complicated and unpredictable post-cured geometries make their design very challenging. Accurate computational models are therefore essential to the development of these materials. The computational framework described in this paper possesses a variety of novel features which have never been tried for this class of composites and can potentially help to fully automatise and improve their design process. A unified approach is used to impose the representative volume element boundary conditions, which allows convenient switching between linear displacement, uniform traction and periodic boundary conditions. The computational framework is implemented using hierarchic basis functions of arbitrary polynomial order, which allows one to increase the order of approximation without changing the finite element mesh. The yarns' principal directions, required for the transversely isotropic material model are calculated using a potential flow analysis along these yarns. This feature is very useful for 3D-textile composites and can accurately determine fibres’ directions even in the case of very deformed yarns. A numerical example from literature consisting of a 3D-orthogonal woven composite is used to demonstrate the correct implementation and performance of the developed computational framework. Also, the developed computational framework is used to perform a comparative study of the homogenised mechanical properties of five 3D-textile composites with different yarn architectures.

Change detection in urban landscapes: a tensor factorization approach

Analysis of urban landscape has been an interesting research challenge for decades. The advent of machine learning and data mining techniques have geared the problem from simple analysis of data to knowledge discovery from data. This work attempts to mine urban landscapes to find the change pattern which has happened over the region for a period of interest. The work proposes a spatiotemporal-metric miner, which uses the spatial, temporal and landscape metric data to discover the change that has occurred in a region. The model works on a hierarchical basis, wherein, the regions of interest are chosen in a landscape and are aggregated to find the change that has happened over the entire region. The entire model is built by taking advantage of the tensorized representation of data, and thus resulting in the effective mining of tensors. The growth of a landscape is evaluated regarding two parameters, namely, Inter-class Growth Index and Intra-class Growth Index. Experiments are performed on the landscape regions of Indian cities, and a ranking of cities is presented based on the growth indices, which are validated against standards. In the experiments, Jaipur city showed the highest Inter-class Growth Index value of 2.68 and Surat city had an Intra-class Growth Index of 0.78.

Hierarchical p‐version C1 finite elements on quadrilateral and triangular domains with curved boundaries and their applications to Kirchhoff plates

SummaryThis work focuses on the construction of p‐version finite elements that have curve boundaries for C1 problems. Both triangular and quadrilateral elements are constructed based on the C1‐version blending function interpolation methods that are developed in this work and in the literature. Orthogonal hierarchical bases are constructed and subsequently transformed into interpolative nodal bases to facilitate the imposition of boundary conditions and the implementation of C1 conformity on curvilinear domains. Nodal collocation strategies are also studied for improving the numerical performance, and novel nonuniformly distributed nodes, namely, Gauss‐Jacobi (GJ) points, are proposed. For parallelograms and straight‐sided triangular elements, C1 continuity is exactly satisfied between neighboring elements. The difficulty of C1 conformity for elements that have curved boundaries is circumvented by interpolating the normal derivatives at Gauss‐Lobatto nodes. Moreover, with the help of the blending function interpolation method, the bases on edges and the internal modes can differ in terms of approximation order. Therefore, local p‐refinements can be easily performed by these elements. Numerical results demonstrated that these elements are computationally inexpensive and converge fast for problems with regular and irregular domains.

Hierarchical Matrix Operations on GPUs

Hierarchical matrices are space- and time-efficient representations of dense matrices that exploit the low-rank structure of matrix blocks at different levels of granularity. The hierarchically low-rank block partitioning produces representations that can be stored and operated on in near-linear complexity instead of the usual polynomial complexity of dense matrices. In this article, we present high-performance implementations of matrix vector multiplication and compression operations for the H 2 variant of hierarchical matrices on GPUs. The H 2 variant exploits, in addition to the hierarchical block partitioning, hierarchical bases for the block representations and results in a scheme that requires only O ( n ) storage and O ( n ) complexity for the mat-vec and compression kernels. These two operations are at the core of algebraic operations for hierarchical matrices, the mat-vec being a ubiquitous operation in numerical algorithms while compression/recompression represents a key building block for other algebraic operations, which require periodic recompression during execution. The difficulties in developing efficient GPU algorithms come primarily from the irregular tree data structures that underlie the hierarchical representations, and the key to performance is to recast the computations on flattened trees in ways that allow batched linear algebra operations to be performed. This requires marshaling the irregularly laid out data in a way that allows them to be used by the batched routines. Marshaling operations only involve pointer arithmetic with no data movement and as a result have minimal overhead. Our numerical results on covariance matrices from 2D and 3D problems from spatial statistics show the high efficiency our routines achieve over 550GB/s for the bandwidth-limited matrix-vector operation and over 850GFLOPS/s in sustained performance for the compression operation on the P100 Pascal GPU.

A new implicit hybridizable discontinuous Galerkin time-domain method for solving the 3-D electromagnetic problems

Based on an implicit time scheme, this paper presents a new implicit hybridizable discontinuous Galerkin time-domain (imHDGTD) method for solving the 3-D electromagnetic problems with locally refined meshes. Compared with the classical discontinuous Galerkin (DG) method, the proposed imHDGTD method has a larger suitable time step and the reduction in number of degrees of freedom (DOFs). Besides, we adopt the nodal hierarchical basis functions and further propose a matrix solving technology with an effective p-type multigrid preconditioner, which decreases significantly imHDGTD’s matrix factorization time and memory usage. Some numerical results show the accuracy and a higher computational performance of the imHDGTD, which illustrates this method more promising and effective for the complex multiscale electromagnetic problem.

Multidimensional Hierarchical Interpolation Method on Sparse Grids for the Absorption Problem

The numerical integration of multidimensional functions using some variables of the sparse grid method for the absorption problem is presented in this paper. The multivariate quadrature expressions are constructed by combining tensor of suited one dimensional formula. We develop a multidimensional adaptive quadrature algorithm for the implementation of sparse grid based on a hierarchical basis. Furthermore, we obtain a new error bound at each sparse grid point. The numerical examples are shown to demonstrate the efficiency of our algorithm for the absorption problem and confirm the theoretical estimates.

Preconditioning the Mass Matrix for High Order Finite Element Approximation on Triangles

The problem of preconditioning the $p$-version mass matrix on meshes of (possibly curvilinear) triangular elements in two dimensions is considered. Through a judicious choice of hierarchical basis,...

BOUSSINESQ MODELING OF SHIP-WAKES AND THEIR CONTRIBUTION TO COASTAL EROSION IN AN ADAPTIVE MESH REFINEMENT SYSTEM

Heavy ship traffic causes a growing concern with respect to public safety, potential damage of coastal structures, and corresponding environmental impacts. High-speed ship-generated wakes such as solitons, undular bores, and breaking bores behave differently compared to wind waves and have a great potential for damage in vulnerable areas such as low-energy coasts and wetlands. In this study, we are developing a multi-grid model framework for the Fully Nonlinear Boussinesq Wave Model, FUNWAVE-TVD (Shi et al., 2012), for simulating ship-wakes in both the operational scale and refined process scales using full two-way coupling. Physical processes in areas of interest requiring higher model resolution, such as the ship-wake generation region, wave breaking in the near-field, and wave evolution with wave-structure interaction in the nearshore field, will be modeled in refined grids embedded in the operational-scale domain. A dynamically adaptive grid algorithm is implemented in order to track a vessel and calculate the physical processes precisely in the wave generation and breaking region in the vicinity of the vessel. Both pressure source and panel source methods for ship wave generation will be tested in the model framework. A concept of nesting layers based on the hierarchical basis, and an efficient parallelization method in the context of the full domain partition are utilized to allow the model to deal with a large-scale computation efficiently in a High Performance Computing (HPC) system.

Development of [formula omitted]-inverse model by using generalized polynomial chaos

We present a hp-inverse model to estimate a smooth, non-negative source function from a limited number of observations for a two-dimensional linear source inversion problem. A standard least-square inverse model is formulated by using a set of Gaussian radial basis functions (GRBF) on a rectangular mesh system with a uniform grid space. Here, the choice of the mesh system is modeled as a random variable and the generalized polynomial chaos (gPC) expansion is used to represent the random mesh system. It is shown that the convolution of gPC and GRBF provides hierarchical basis functions for the linear source inverse model with the hp-refinement capability. We propose a mixed l1 and l2 regularization to exploit the hierarchical nature of the basis functions to find a sparse solution. The hp-inverse model has an advantage over the standard least-square inverse model when the number of data is limited. It is shown that the hp-inverse model provides a good estimate of the source function even when the number of unknown parameters (m) is much larger the number of data (n), e.g., m∕n>40.

An adaptive isogeometric analysis collocation method with a recovery-based error estimator

In this paper, we propose an enhanced isogeometric analysis (IGA) collocation method. It is well known that the location of the collocation points plays an important role in the accuracy and stability of IGA collocation methods. This is particularly true for non-uniform meshes and domains generated from multi-patch geometries. We present an enhanced collocation method based on Gauss points, which has improved accuracy as compared to using C1 splines and a recovery-based error estimator that can be derived by sampling the solution at particular points in the domain. Adaptivity is implemented using a hierarchical spline basis, which satisfies the C1 continuity requirement. The proposed approach has been tested by several benchmark problems, including multipatch domains and geometries with re-entrant corners.

Hierarchical Divergence Conforming Bases for Edge Singularities in Quadrilateral Cells

Singular divergence-conforming bases have been proposed for the solution of integral equations although they have seen only occasional use in practical applications. The existing singular bases are not hierarchical, which prevents their use in adaptive p-refinement applications. In this paper, a new family of singular hierarchical basis functions is proposed for quadrilateral cells. These functions model the singularities associated with current and charge density at edges and are more convenient for modeling such singularities than triangular bases of the same kind. The basis functions are of the additive kind and combine a hierarchical polynomial representation on quadrilaterals with linearly independent singular terms that incorporate general exponents that may be adjusted for the specific wedge angle of interest. Moreover, the added singular basis functions are computed on the fly. On the basis of various reported numerical results, this paper also illustrates the difficulties, the advantages, the accuracy, and the cost of using such bases in the method of moment solutions of integral equations.

Three-dimensional thermo-mechanical solutions of cross-ply laminated plates and shells by a differential quadrature hierarchical finite element method

The thermo-elastic behaviour induced by temperature in composite structures has become a significant factor in structural design. In this paper, a three-dimensional thermo-elastic theory for composite cross-ply laminated plates was discretized by a differential quadrature hierarchical finite element method (DQHFEM). The DQHFEM is a weak-form differential quadrature method using hierarchical bases that can provide highly accurate results using only a few sampling points. Flexible and efficient wedge (triangular prism) and hexahedron elements were constructed and applied to three-dimensional (3D) analyses of cross-ply laminated plates and shells for the first time. Compared with various models in literatures, the DQHFEM showed very good agreements with exact solutions based on 3D elastic theory. The thermo-mechanical analyses of cross-ply laminated plates and shells based on 3D theory indicate that the DQHFEM is an effective method for high accuracy analyses with low computational costs, because the DQHFEM can provide results with high accuracy by using only several nodes on the thickness direction.

A high-order method for simulating convective planar Poiseuille flow over a heated rotating sphere

PurposeThis paper aims to present a high-order algorithm implemented with the modal spectral element method and simulations of three-dimensional thermal convective flows by using the full viscous dissipation function in the energy equation. Three benchmark problems were solved to validate the algorithm with exact or theoretical solutions. The heated rotating sphere at different temperatures inside a cold planar Poiseuille flow was simulated parametrically at varied angular velocities with positive and negative rotations.Design/methodology/approachThe fourth-order stiffly stable schemes were implemented and tested for time integration. To provide the hp-refinement and spatial resolution enhancement, a modal spectral element method using hierarchical basis functions was used to solve governing equations in a three-dimensional space.FindingsIt was found that the direction of rotation of the heated sphere has totally different effects on drag, lateral force and torque evaluated on surfaces of the sphere and walls. It was further concluded that the angular velocity of the heated sphere has more influence on the wall normal velocity gradient than on the wall normal temperature gradients and therefore, more influence on the viscous dissipation than on the thermal dissipation.Research limitations/implicationsThis paper concerns incompressible fluid flow at constant properties with up to medium temperature variations in the absence of thermal radiation and ignoring the pressure work.Practical implicationsThis paper contributes a viable high-order algorithm in time and space for modeling convective heat transfer involving an internal heated rotating sphere with the effect of viscous heating.Social implicationsResults of this paper could provide reference for related topics such as enhanced heat transfer forced convection involving rotating spheres and viscous thermal effect.Originality/valueThe merits include resolving viscous dissipation and thermal diffusion in stationary and rotating boundary layers with both h- and p-type refinements, visualizing the viscous heating effect with the full viscous dissipation function in the energy equation and modeling the forced advection around a rotating sphere with varied positive and negative angular velocities subject to a shear flow.

A two-stage ensemble Kalman filter based on multiscale model reduction for inverse problems in time fractional diffusion-wave equations

Ensemble Kalman filter (EnKF) has been widely used in state estimation and parameter estimation for the dynamic system where observational data is obtained sequentially in time. Very burdened simulations for the forward problem are needed to update a large number of EnKF ensemble samples. This will slow down the analysis efficiency of the EnKF for large-scale and high dimensional models. To reduce uncertainty and accelerate posterior inference, a two-stage ensemble Kalman filter is presented to improve the sequential analysis of EnKF. It is known that the final posterior ensemble may be concentrated in a small portion of the entire support of the initial prior ensemble. It will be much more efficient if we first build a new prior by some partial observations, and construct a surrogate only over the significant region of the new prior. To this end, we construct a very coarse model using generalized multiscale finite element method (GMsFEM) and generate a new prior ensemble in the first stage. GMsFEM provides a set of hierarchical multiscale basis functions supported in coarse blocks. This gives flexibility and adaptivity to choosing degree of freedoms to construct a reduce model. In the second stage, we build an initial surrogate model based on the new prior by using GMsFEM and sparse generalized polynomial chaos (gPC)-based stochastic collocation methods. To improve the initial surrogate model, we dynamically update the surrogate model, which is adapted to the sequential availability of data and the updated analysis. The two-stage EnKF can achieve a better estimation than standard EnKF, and significantly improve the efficiency to update the ensemble analysis (posterior exploration). To enhance the applicability and flexibility in Bayesian inverse problems, we extend the two-stage EnKF to non-Gaussian models and hierarchical models. In the paper, we focus on the time fractional diffusion-wave models in porous media and investigate their Bayesian inverse problems using the proposed two-stage EnKF. A few numerical examples are carried out to demonstrate the performance of the two-stage EnKF method by taking account of parameter and structure inversion in permeability fields and source functions.

A Haar wavelet method for angularly discretising the Boltzmann transport equation

A novel, hierarchical Haar wavelet basis is introduced and used to discretise the angular dimension of the Boltzmann transport equation. This is used in conjunction with a finite element subgrid scale method. This combination is then validated using two steady-state radiation transport problems, namely a 2D dogleg-duct shielding problem and the 2D C5MOX OECD/NEA benchmark. It is shown that the scheme has many similarities to a traditional equal weighted discrete ordinates (Sn) angular discretisation, but the strong motivation for our hierarchical Haar wavelet method is the potential for adapting in angle in a simple fashion through elimination of redundant wavelets. Initial investigations of this adaptive approach are presented for a shielding and criticality eigenvalue example. It is shown that a 60% reduction in the number of angles needed on most spatial nodes - and rising up to 90% on nodes located in high streaming areas - can be attained without adversely affecting the accuracy of the solution.

Adaptive refinement of hierarchical T-splines

We present an adaptive local refinement technique for isogeometric analysis based on hierarchical T-splines. An element-wise point of view is adopted, which exploits Bézier extraction, and allows adaptive refinement of standard hierarchical T-splines and truncated hierarchical T-splines in a straightforward and unified manner. No explicit basis function operations are required to build the hierarchical basis function space, as only matrix manipulations are involved. This makes the efficiency superior to that of existing implementations. In particular, the implementation of truncated hierarchical T-splines requires no explicit truncation of the basis functions. In the analysis, a multi-level T-mesh is constructed by successive cell subdivisions of an initial, coarse T-mesh. An important feature is that Bézier extraction is employed to compute the refinement operator between two successive hierarchical levels, and that, at each level, Bézier extraction is applied to obtain the stiffness matrix without, initially, considering multi-level interaction. This interaction is recovered through a subdivision operator. Numerical examples are presented for validation purposes, and to assess the convergence properties.

Hierarchical loop‐flower basis for solving electric field integral equation

A novel kind of hierarchical loop-flower bases is presented for solving electric field integral equation. Compared with conventional hierarchical loop-tree-based method, the total bases for proposed hierarchical loop-flower-based method can be reduced a third. Thus, the memory and time cost can be largely reduced. The hierarchical loop-flower basis function is defined on a hierarchical mesh which can be generated by a mesh refinement algorithm. By expanding the current with the hierarchical loop-flower bases, the solenoidal and non-solenoidal current are separated and thus the EFIE low-frequency breakdown issue can be remedied. In addition to that a multiresolution preconditioner technology is also introduced to fast up the iterative convergence speed. Some examples show that the basis number is largely reduced.

Quantifying the Impact of Random Surface Perturbations on Reflective Gratings

We present a novel deterministic method capable of calculating statistical moments of transverse electric polarized fields scattered by perfect electric conductor gratings with small surface random perturbations. Based on a first-order shape Taylor expansion, the resulting electric field integral equations are solved via the method of moments with constant hierarchical basis or Haar wavelets. This allows for a sparse tensor approximation, significantly reducing the number of required unknowns and yielding a higher rate of convergence than a dense approximation. Moreover, the proposed approach converges faster than Monte Carlo simulations with significantly less computational effort. Validation of the proposed approach is performed for several cases, and realistic simulations applied to the calculation and prediction of grating efficiency reveal the applicability of the method.

Modal spectral element method in curvilinear domains

We present a high-order spectral element method (SEM) using modal or hierarchical basis for modeling of the two-dimensional linear and non-linear partial differential equations in complex geometry. The discretization is based on conforming spectral element in space and finite difference methods in time. Unlike the nodal SEM which is based on the Lagrange polynomials associated with the Gauss–Lobatto–Legendre or Chebyshev quadrature nodes, the Lobatto polynomials are used in this paper as modal basis. Using modal bases, due to their orthogonal properties, enables us to obtain the elemental matrices efficiently in the linear problems. The difficulty of implementation of modal approximations for non-linear problems is treated in this paper by expanding the non-linear terms in the weak form of differential equations in terms of the Lobatto polynomials on each element using the fast Fourier transform (FFT) algorithm. Utilization of Fourier interpolation on equidistant points in the FFT algorithm can be suitable at minimizing aliasing error. On the other hand, the polynomial order of approximations for non-linear terms can be sufficiently large to capture major variations and render the aliasing error so small. The proposed approach also leads to finding numerical solution of a non-linear differential equation through solving a system of linear algebraic equations. The non-constant Jacobian, as well as any non-polynomial coefficients and nonlinearities arisen in the entries of mass and stiffness matrices are expanded in terms of the Legendre polynomials by FFT algorithm and so they can be produced in the complex geometry accurately even in a tensor product form. Efficiency and exponential convergence properties of the approximation scheme are validated through several non-trivial examples in the numerical results section.

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.