Modified Basis Functions Research Articles

Polynomial Approximation over Arbitrary Shape Domains

In spectral/finite element methods, a robust and stable high-order polynomial approximation method for the solution can significantly reduce the required number of degrees of freedom (DOFs) to achieve a certain level of accuracy. In this work, a closed-form relation is proposed to approximate the Fekete points (AFPs) on arbitrary shape domains based on the singular value decomposition (SVD) of the Vandermonde matrix. In addition, a novel method is derived to compute the moments on highly complex domains, which may include discontinuities. Then, AFPs are used to generate compatible basis functions using SVD. Equations are derived and presented to determine orthogonal/orthonormal modal basis functions, as well as the Lagrange basis. Furthermore, theorems are proved to show the convergence and accuracy of the proposed method, together with an explicit form of the Weierstrass theorem for polynomial approximation. The method was implemented and some classical cases were analyzed. The results show the superior performance of the proposed method in terms of convergence and accuracy using many fewer DOFs and, thus, a much lower computational cost. It was shown that the orthogonal modal basis is the best choice to decrease the DOFs while maintaining a small Lebesgue constant when very high degree of polynomial is employed.

Gaussian expansion element method of the new dynamic modeling technique in non-uniform and variable cross-section structures

The non-uniform and variable cross-section structures are widely used in engineering. However, its non-uniform characteristic has also caused some troubles in existing dynamic modeling methods, such as poor convergence, accuracy, efficiency, and difficulty in predicting high-order derivatives of mode shape. To solve these problems, this paper provides a new dynamic modeling technology for non-uniform structures based on element division and the Gaussian Expansion Method (GEM), named Gaussian Expansion Element Method (GEEM). In the GEEM, the non-uniform structure is divided into several elements, and their displacement fields are expanded to a set of Gaussian functions. For ensuring every element meets boundary and continuity conditions, these Gaussian basis functions are divided into modified (used to modify displacement fields near end supports) and main basis functions (describing internal displacement fields of element). Then, the dynamic model of element can be derived by the Rayleigh-Ritz method, and its state variables include the generalized coordinates of modified and main basis functions. Finally, the dynamic equation of the non-uniform structure can be obtained by assembling these equations of elements and eliminating the coordinates of modified basis functions by using boundary and continuity conditions. Numerical results show that the new method can predict the dynamic behaviors of the non-uniform beam, periodic structure, and frame structure, calculate their high-order derivatives of mode shape, and has higher accuracy, efficiency, and numerical stability than GEM. Notably, an adaptive selection scheme of Gaussian basis functions and element division rules are given to further enhance the new method's accuracy and efficiency.

Asymptotic coefficients and errors for Chebyshev polynomial approximations with weak endpoint singularities: Effects of different bases

When one solves differential equations by a spectral method, it is often convenient to shift from Chebyshev polynomials Tn(x) with coefficients an to modified basis functions that incorporate the boundary conditions. For homogeneous Dirichlet boundary conditions, u(±1) = 0, popular choices include the “Chebyshev difference basis” ςn(x) ≡ Tn+2(x)−Tn(x) with coefficients here denoted by bn and the “quadratic factor basis” ϱn(x) ≡ (1 − x2)Tn(x) with coefficients cn. If u(x) is weakly singular at the boundary, then the coefficients an decrease proportionally to \({\cal O}\left( {A\left( n \right)/{n^\kappa }} \right)\) for some positive constant κ, where A(n) is a logarithm or a constant. We prove that the Chebyshev difference coefficients bn decrease more slowly by a factor of 1/n while the quadratic factor coefficients cn decrease more slowly still as \({\cal O}\left( {A\left( n \right)/{n^{\kappa - 2}}} \right)\). The error for the unconstrained Chebyshev series, truncated at degree n = N, is \({\cal O}\left( {\left| {A\left( N \right)} \right|/{N^\kappa }} \right)\) in the interior, but is worse by one power of N in narrow boundary layers near each of the endpoints. Despite having nearly identical error norms in interpolation, the error in the Chebyshev basis is concentrated in boundary layers near both endpoints, whereas the error in the quadratic factor and difference basis sets is nearly uniformly oscillating over the entire interval in x. Meanwhile, for Chebyshev polynomials, the values of their derivatives at the endpoints are \({\cal O}\left( {{N^2}} \right)\), but only \({\cal O}\left( N \right)\) for the difference basis. Furthermore, we give the asymptotic coefficients and rigorous error estimates of the approximations in these three bases, solved by the least squares method. We also find an interesting fact that on the face of it, the aliasing error is regarded as a bad thing; actually, the error norm associated with the downward curving spectral coefficients decreases even faster than the error norm of infinite truncation. But the premise is under the same basis, and when involving different bases, it may not be established yet.

Efficient spline regression for neural spiking data.

Point process generalized linear models (GLMs) provide a powerful tool for characterizing the coding properties of neural populations. Spline basis functions are often used in point process GLMs, when the relationship between the spiking and driving signals are nonlinear, but common choices for the structure of these spline bases often lead to loss of statistical power and numerical instability when the signals that influence spiking are bounded above or below. In particular, history dependent spike train models often suffer these issues at times immediately following a previous spike. This can make inferences related to refractoriness and bursting activity more challenging. Here, we propose a modified set of spline basis functions that assumes a flat derivative at the endpoints and show that this limits the uncertainty and numerical issues associated with cardinal splines. We illustrate the application of this modified basis to the problem of simultaneously estimating the place field and history dependent properties of a set of neurons from the CA1 region of rat hippocampus, and compare it with the other commonly used basis functions. We have made code available in MATLAB to implement spike train regression using these modified basis functions.

An improved model along with a spectral numerical simulation for fractional predator–prey interactions

Predator–prey models appear in various fields of bio-mathematics used for the analysis of interactions of biological systems. Due to the complexities of the physical context for the real-world problems of food chain dynamics, introducing new models compatible with experimental results stays ongoing research. Many models have been proposed and analyzed for these systems in recent years. In this paper, we propose a new fractional-order predator–prey model with negative feedback on both species with memory-dependent effects, which increases the compatibility level of the model. Then we present a novel Laguerre spectral numerical simulation for the proposed model by introducing Laguerre modal basis functions with collocation and Galerkin techniques. We then transfer the nonlinear model into a system of algebraic equations, which is solved by efficient numerical solvers. Finally, we provide some test problems to show the efficiency of the proposed model and the computational method.

Modified basis functions for MPHT-splines

By allowing split-in-half in mesh refinement, MPHT-splines (modified polynomial splines over hierarchical T-meshes) were recently introduced to increase the flexibility of PHT-splines. However, some basis functions of MPHT-splines may severely decay as the basis functions of PHT-splines during the process of certain refinement of hierarchical T-meshes. This decay phenomenon may cause the isogeometric analysis matrices assembled by these basis functions to be ill-conditioned. To overcome this defect, we present a method to modify the basis functions of MPHT-splines when the supports of the original truncated basis functions are rectangular domains. Hence, the truncation operations are less used, which alleviates the decay phenomenon of MPHT-splines. In addition, the modified basis functions preserve the good properties of the original MPHT-spline basis functions, e.g., the partition of unity, local support and linear independence. Numerical examples show that the condition numbers of isogeometric analysis matrices significantly decrease when new basis functions are applied.

Image analysis using modified exponent-Fourier moments

Classic exponent-Fourier moments (EFMs) have been popularly used for image reconstruction and invariant classification. However, EFMs lack natively the translation and scaling-invariant; in addition, they exhibit two types of drawbacks, namely numerical instability and reconstruction error, which in turn influence their reconstruction capability and image classification accuracy. This study considers the challenge of defining modified EFMs (MEFMs), which are based on modified exponent polynomials. In our methods, the basis function of traditional EFMs is appropriately modified, and these modified basis functions are used to replace the original ones. The basis function of the proposed moments is composed of piecewise modified exponent polynomials modulated by a variable parameter exponential envelope. Various types of optimal-order moments can be established by slightly adjusting the bandwidth of the modified basis functions. Finally, we extend the rotation-invariant feature of previous works and propose a new method of scaling and rotation-invariant image recognition using the proposed moments in a log-polar coordinate domain. The translation invariance can then be achieved by an image projection operation, which is substituted for the traditional approach based on the calculation of image geometric moments. The experimental results demonstrate that the MEFMs perform better than traditional EFMs and other classic orthogonal moments including the latest image moments in terms of the image reconstruction capability and the invariant recognition accuracy of smoothing filters, in both noise-free and noisy conditions.

Discontinuous Galerkin formulation for 2D hydrodynamic modelling: Trade-offs between theoretical complexity and practical convenience

In the modelling of hydrodynamics, the Discontinuous Galerkin (DG) approach constitutes a more complex and modern alternative to the well-established finite volume method. The latter retains some desired practical features for modelling hydrodynamics, such as well-balancing between spatial flux and steep topography gradients, ability to incorporate wetting and drying processes, and computational affordability. In this context, DG methods were originally devised to solve the two-dimensional (2D) Shallow Water Equations (SWE) with irregular topographies and wetting and drying, albeit at reduction in the formulation’s complexity to often being second-order accurate (DG2). The aims of this paper are: (a) to outline a so-called “slope-decoupled” formulation of a standard 2D-DG2-SWE simulator in which theoretical complexity is deliberately reduced; (b) to highlight the capabilities of the proposed slope-decoupled simulator in providing a setting where the simplifying assumptions are verified within the formulation. Both the standard and the slope-decoupled 2D-DG2-SWE models adopt 2D modal basis functions for shaping local planar DG2 solutions on quadrilateral elements, by using an average coefficient and two slope coefficients along the Cartesian coordinates. Over a quadrilateral element, the stencil of the slope-decoupled 2D-DG2 formulation is simplified to remove the interdependence of slope-coefficients for both flow and topography approximations. The fully well-balanced character the slope-decoupled 2D-DG2-SWE planar solutions is theoretically studied. The performance of the latter is compared with the standard 2D-DG2 formulation in classical simulation tests. Other tests are conducted to diagnostically verify the conservative properties of the 2D-DG2-SWE method in scenarios involving sharp topography gradients and wet and/or dry zones. The analyses conducted offer strong evidence that the proposed slope-decoupled 2D-DG2-SWE simulator is very attractive for the development of robust flood models.

Projection methods for stochastic dynamic systems: A frequency domain approach

A collection of hybrid projection approaches are proposed for approximating the response of stochastic partial differential equations which describe structural dynamic systems. In this study, an optimal basis for the approximation of the response of a stochastically parametrised structural dynamic system has been computed from its generalised eigenmodes. By applying appropriate approximations in conjunction with a reduced set of modal basis functions, a collection of hybrid projection methods are obtained. These methods have been further improved by the implementation of a sample based Galerkin error minimisation approach. In total six methods are presented and compared for numerical accuracy and computational efficiency. Expressions for the lower order statistical moments of the hybrid projection methods have been derived and discussed. The proposed methods have been implemented to solve two numerical examples: the bending of a Euler–Bernoulli cantilever beam and the bending of a Kirchhoff–Love plate where both structures have stochastic elastic parameters. The response and accuracy of the proposed methods are subsequently discussed and compared with the benchmark solution obtained using an expensive Monte Carlo method.

A Memory Saving Augmented EFIE With Modified Basis Functions for Low-Frequency Problems

A new augmented electric field integral equation (A-EFIE) formulation is discussed in the low frequency. In the new formulation, both the electric current and charge are considered as unknowns. Different from the conventional A-EFIE, a constraint on the normal component of the magnetic field is added to modifying the formulation of A-EFIE for improving the accuracy and a kind of constant nodal functions is introduced for expanding the electric charge. With the proposed method, besides avoiding the low-frequency breakdown, the storage of impedance matrix is reduced over 35% compared with the conventional A-EFIE, which is owing to the application of constant nodal functions associated with vertices. Numerical examples illuminate the validity and performance of the new A-EFIE with constant nodal functions.

Implicit-Modal Discontinuous Galerkin Scheme for Two-Phase Flow with Discontinuous Capillary Pressure

In this paper, we propose an implicit-modal discontinuous Galerkin scheme for solving an incompressible and immiscible oil-water flow through a heterogeneous reservoir with different rock types. The scheme uses the modal discontinuous Galerkin method for spatial discretization of both pressure and saturation equations and the backward Euler method for temporal discretization. In the modal discontinuous Galerkin method, we apply the well-localized bases that are constructed by both the nodal and modal basis functions. To find the approximate solution for the saturation equation, we propose the upwind technique. In the stability analysis of the scheme, we obtain the constant of the trace inequality for the approximate function and its first derivatives using the modal basis functions. In this way, we give suitable ranges for penalty terms. To guarantee the convergence of the approximate solution, we find suitable ranges of the time step. To demonstrate the applicability of the proposed scheme, we apply it to the dense nonaqueous phase liquids and the Versuchseinrichtung zur Grundwasser und Altlastensanierung infiltration problems. Numerical results are presented.

Nonintrusive Uncertainty Analysis of Fluid-Structure Interaction with Spatially Adaptive Sparse Grids and Polynomial Chaos Expansion

The propagation of uncertainty in physical parameters of fluid-structure interaction problems is a challenging task---both mathematically and in terms of computational workload. In this paper, we employ nonintrusive polynomial chaos expansion and model the uncertainty in five independent input parameters that characterize both fluid and structure. We propose a novel methodology to compute the expansion's coefficients using spatially adaptive sparse grids and products of one-dimensional integrals, which exploits the tensor structure of both sparse grids and probabilistic space. Furthermore, with spatial adaptivity and modified basis functions, we keep the number of sparse grid points small. We test our approach in two test cases: (i) an elastic vertical flap in a channel flow and (ii) a computationally challenging, well-established benchmark. The outputs of interest are the x-deflection and total force on the structure. In the first test case, we consider six implementations of our methodology and two established methods based on sparse grid quadrature. We evaluate the performance (in terms of number of runs of the numerical solver) and accuracy of all eight methods. The results show that one variant of our approach outperforms all the other implementations. We apply our best variant to the benchmark scenario and address the high computational demands of the resulting problem using three levels of parallelism. Importantly, our approach is not restricted to fluid-structure interaction problems. We can address a broad spectrum of computationally expensive problems, provided that sparse grid approximations can be employed.

Modified Bases of PHT-Splines

Recently, it was found that during the process of certain refinement of hierarchical T-meshes, some basis functions of PHT-splines decay severely, which is not expected in solving numerical PDEs and in least square data fitting since the matrices assembled by these basis functions are likely to be ill-conditioned. In this paper, we present a method to modify the basis functions of PHT-splines in the case that the supports of the original truncated basis functions are rectangular domains to overcome the decay problem. The modified basis functions preserve the same nice properties of the original PHT-spline basis functions such as partition of unity, local support, linear independency. Numerical examples show that the modified basis functions can greatly decrease the condition numbers of the stiffness matrices assembled in solving Poisson’s equation with Dirichlet boundary conditions.

Bernstein modal basis: Application to the spectral Petrov‐Galerkin method for fractional partial differential equations

In the spectral Petrov‐Galerkin methods, the trial and test functions are required to satisfy particular boundary conditions. By a suitable linear combination of orthogonal polynomials, a basis, that is called the modal basis, is obtained. In this paper, we extend this idea to the nonorthogonal dual Bernstein polynomials. A compact general formula is derived for the modal basis functions based on dual Bernstein polynomials. Then, we present a Bernstein‐spectral Petrov‐Galerkin method for a class of time fractional partial differential equations with Caputo derivative. It is shown that the method leads to banded sparse linear systems for problems with constant coefficients. Some numerical examples are provided to show the efficiency and the spectral accuracy of the method.

Seasonal low-degree changes in terrestrial water mass load from global GNSS measurements

Large-scale mass redistribution in the terrestrial water storage (TWS) leads to changes in the low-degree spherical harmonic coefficients of the Earth’s surface mass density field. Studying these low-degree fluctuations is an important task that contributes to our understanding of continental hydrology. In this study, we use global GNSS measurements of vertical and horizontal crustal displacements that we correct for atmospheric and oceanic effects, and use a set of modified basis functions similar to Clarke et al. (Geophys J Int 171:1–10, 2007) to perform an inversion of the corrected measurements in order to recover changes in the coefficients of degree-0 (hydrological mass change), degree-1 (centre of mass shift) and degree-2 (flattening of the Earth) caused by variations in the TWS over the period January 2003–January 2015. We infer from the GNSS-derived degree-0 estimate an annual variation in total continental water mass with an amplitude of $$(3.49 \pm 0.19) \times 10^{3}$$ Gt and a phase of $$70^{\circ } \pm 3^{\circ }$$ (implying a peak in early March), in excellent agreement with corresponding values derived from the Global Land Data Assimilation System (GLDAS) water storage model that amount to $$(3.39 \pm 0.10) \times 10^{3}$$ Gt and $$71^{\circ } \pm 2^{\circ }$$ , respectively. The degree-1 coefficients we recover from GNSS predict annual geocentre motion (i.e. the offset change between the centre of common mass and the centre of figure) caused by changes in TWS with amplitudes of $$0.69 \pm 0.07$$ mm for GX, $$1.31 \pm 0.08$$ mm for GY and $$2.60 \pm 0.13$$ mm for GZ. These values agree with GLDAS and estimates obtained from the combination of GRACE and the output of an ocean model using the approach of Swenson et al. (J Geophys Res 113(B8), 2008) at the level of about 0.5, 0.3 and 0.9 mm for GX, GY and GZ, respectively. Corresponding degree-1 coefficients from SLR, however, generally show higher variability and predict larger amplitudes for GX and GZ. The results we obtain for the degree-2 coefficients from GNSS are slightly mixed, and the level of agreement with the other sources heavily depends on the individual coefficient being investigated. The best agreement is observed for $$T_{20}^C$$ and $$T_{22}^S$$ , which contain the most prominent annual signals among the degree-2 coefficients, with amplitudes amounting to $$(5.47 \pm 0.44) \times 10^{-3}$$ and $$(4.52 \pm 0.31) \times 10^{-3}$$ m of equivalent water height (EWH), respectively, as inferred from GNSS. Corresponding agreement with values from SLR and GRACE is at the level of or better than $$0.4 \times 10^{-3}$$ and $$0.9 \times 10^{-3}$$ m of EWH for $$T_{20}^C$$ and $$T_{22}^S$$ , respectively, while for both coefficients, GLDAS predicts smaller amplitudes. Somewhat lower agreement is obtained for the order-1 coefficients, $$T_{21}^C$$ and $$T_{21}^S$$ , while our GNSS inversion seems unable to reliably recover $$T_{22}^C$$ . For all the coefficients we consider, the GNSS-derived estimates from the modified inversion approach are more consistent with the solutions from the other sources than corresponding estimates obtained from an unconstrained standard inversion.

Hierarchical model reduction of nonlinear partial differential equations based on the adaptive empirical projection method and reduced basis techniques

In this paper we extend the hierarchical model reduction framework based on reduced basis techniques for the application to nonlinear partial differential equations. The major new ingredient to accomplish this goal is the introduction of the adaptive empirical projection method, which is an adaptive integration algorithm based on the (generalized) empirical interpolation method. Different from other partitioning concepts for the empirical interpolation method we perform an adaptive decomposition of the spatial domain. We project both the variational formulation and the range of the nonlinear operator onto reduced spaces. Those reduced spaces combine the full dimensional (finite element) space in an identified dominant spatial direction and a reduction space or collateral basis space spanned by modal orthonormal basis functions in the transverse direction. Both the reduction and the collateral basis space are constructed in a highly nonlinear fashion by introducing a parametrized problem in the transverse direction and associated parametrized operator evaluations, and by applying reduced basis methods to select the bases from the corresponding snapshots. Rigorous a priori and a posteriori error estimators, which do not require additional regularity of the nonlinear operator are proven for the adaptive empirical projection method and then used to derive a rigorous a posteriori error estimator for the resulting hierarchical model reduction approach. Numerical experiments for an elliptic nonlinear diffusion equation demonstrate a fast convergence of the proposed dimensionally reduced approximation to the solution of the full-dimensional problem. Runtime experiments verify a close to linear scaling of the reduction method in the number of degrees of freedom used for the computations in the dominant direction.

Partition of unity isogeometric analysis of two dimensional elliptic singular perturbation problems

The design basis functions on the reference domain in IGA are diversified and enhanced by extra enrichment functions and various local refinements with the use of partition of unity (PU) function with flat-top. These reconditioned and modified basis functions are pushed forward to the physical domain by the original design mapping for analysis. With this method, the corresponding stiffness matrix has a small bandwidth and local refinement is simple. Moreover, we construct the PU functions in the reference domain and then move them to a physical domain through a geometric mapping to be used for the generation of global basis functions on a physical domain. Therefore, we also have several advantages in calculating stiffness matrices and load vectors. Here we apply this method to various boundary layer problems.

Solving Scattering From Multiple Bodies of Revolution by Modal Characteristic Basis Function Method With Sparse Matrix Technique

Modal Characteristic Basis Function method for bodies of revolution (BoR-CBFM) is a fast solver for scattering from multiple BoRs. By using of BoR-CBFM, the modal characteristic basis functions (MCBFs) are constructed, and the size of impedance matrix is greatly reduced. However, the matrix filling, especially the filling of mutual-interaction matrix, is still a huge burden for numerical computation. To further enhance the efficiency of BoR-CBFM and reduce the cost of matrix filling, we present a sparse technique for BoR-CBFM. Through this technique, those weak elements in mutual-interaction matrix that contribute little to the linear system are neglected, and a sparse impedance matrix is achieved. As a result, this technique leads to a reduction on the cost of matrix filling. Numerical results are presented to prove the efficiency of this technique.

FDFD Nonconformal Domain Decomposition Method for the Electromagnetic Modeling of Interconnections in Silicon Interposer

This paper proposes an efficient method to model interconnections with through-silicon vias (TSVs) in silicon interposer for 3-D systems. The proposed method uses 3-D finite-difference frequency-domain nonconformal domain decomposition method to model the redistribution layer transmission lines on lossy silicon interposer. Using the nonconformal domain decomposition approach, the interposer can be divided into separate subdomains. Individual subdomains can be discretized independently based on its feature size. Field continuity at interfaces between domains is enforced by introducing Lagrange multiplier and vector basis functions at the interfaces. TSVs are modeled using an integral-equation-based solver, which uses cylindrical modal basis functions. The formulation on incorporating multiport network into nonconformal domain decomposition is presented to include the parasitic effects of TSV arrays into the system matrix. By comparing with 3-D full-wave simulations and commercial solvers, this paper validates the accuracy and efficiency of the proposed modeling approach.

Modeling of Through-Silicon Via (TSV) Interposer Considering Depletion Capacitance and Substrate Layer Thickness Effects

To support the recent progress in 3-D integration based on through-silicon via (TSV) technology, an improved electromagnetic modeling method for TSVs is presented. In the framework of the mixed-potential integral equations combined with cylindrical modal basis functions, the proposed method can extract the effects of depletion capacitances and a finite substrate. To include the effects of depletion region generated by an external dc bias voltage, an additional capacitive cell is employed around a TSV. The proposed method also considers the effect from the finite silicon substrate accurately by employing the multilayered Green’s functions. To reduce the computational cost for calculations involving Green’s functions, a method to approximate Green’s functions over localized intervals when computing partial potential coefficients is presented. The proposed method is validated for simple TSV examples and shows an improved accuracy with the acceptable usage of memory and simulation time. In addition, a $10 \times 10$ TSV array is modeled using different design parameters, showing the capability for dealing with larger size problems using this method.