•We present an efficient reduced basis method for nonlinear and nonaffine steady state or transient PDEs.•A new and efficiently implementable error estimator based on residual hyper reductions drives ...a greedy algorithm.•The method is analyzed and tested for the viscous Burgers' equation and nonlinear reaction diffusion equations.•The approach can avoid the online efficiency degradation induced by the application of the Empirical Interpolation Method.
The need for multiple interactive, real-time simulations using different parameter values has driven the design of fast numerical algorithms with certifiable accuracies. The reduced basis method (RBM) presents itself as such an option. RBM features a mathematically rigorous error estimator which drives the construction of a low-dimensional subspace. A surrogate solution is then sought in this low-dimensional space approximating the parameter-induced high fidelity solution manifold. However when the system is nonlinear or its parameter dependence nonaffine, this efficiency gain degrades tremendously, an inherent drawback of the application of the empirical interpolation method (EIM).
In this paper, we augment and extend the EIM approach as a direct solver, as opposed to an assistant, for solving nonlinear partial differential equations on the reduced level. The resulting method, called Reduced Over-Collocation method (ROC), is stable and capable of avoiding the efficiency degradation. Two critical ingredients of the scheme are collocation at about twice as many locations as the number of basis elements for the reduced approximation space, and an efficient error indicator for the strategic building of the reduced solution space. The latter, the main contribution of this paper, results from an adaptive hyper reduction of the residuals for the reduced solution. Together, these two ingredients render the proposed R2-ROC scheme both offline- and online-efficient. A distinctive feature is that the efficiency degradation appearing in traditional RBM approaches that utilize EIM for nonlinear and nonaffine problems is circumvented, both in the offline and online stages. Numerical tests on different families of time-dependent and steady-state nonlinear problems demonstrate the high efficiency and accuracy of our R2-ROC and its superior stability performance.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
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.
•Isogeometric collocation at Gauss points leads to an optimally convergent scheme.•A hybrid Galerkin-collocation method is used for the patch and Neumann boundaries.•Adaptive refinement with PHT splines is driven by a recovery-based error estimator.
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•A novel localized collocation Trefftz method (LCTM) with cylindrical T-complete bases is presented for potential-based inverse electromyography (PIE).•The proposed LCTM scheme performs more stable ...than the traditional CTM with a large number of discretization nodes.•The proposed LCTM scheme is accurate and stable for reconstructing the electric potential even with 5% noisy boundary data.•The morphology of LCTM reconstructed electromyography is similar to that of measured electromyography in sheep’s uterine contractions.
This paper introduces a novel localized collocation Trefftz method (LCTM) for potential-based inverse electromyography (PIE). PIE is a noninvasive technique to calculate the internal electrical potentials from measured body surface electromyographic data, which can be considered as an inverse Cauchy problem with potential equation. In the proposed LCTM, the electrical potential at every node is expressed as a linear combination of 3D Trefftz basis in each stencil support, and the sparse linear system is yield by satisfying governing equation at interior nodes and boundary conditions at boundary nodes. The proposed LCTM inherits the properties of easy-to-use and meshless from the collocation Trefftz method (CTM), and mitigates the ill-conditioning resultant matrix encountered in the CTM. Numerical efficiency of the proposed method is investigated in comparison with the CTM and experimental data.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Droughts have a detrimental effect on plenty of social and economic endeavors along with surface and groundwater resources. Therefore, drought must be adequately considered when planning and ...regulating the water supply. This study will look at the latest developments in merging techniques to lessen the inconsistent drought monitoring and characterization attributed to the global standard shortage. The current research considers the framework of three distinct standardized indicators, SPI, SPTI, and SPEI, of six metrological stations in Pakistan from 1971 to 2017, intending to analyze drought using integrating techniques. Two merging techniques, Modified Triple Collocation (MTC) and proposed Scaled Triple Collocation (STC), are examined relative to Triple Collocation (TC). Correlation, Sen’s Slope, Taylor diagram, Kling Gupta Efficiency (KGE), and error variance analyses were used to evaluate their performance. The correlation study reveals that individual series have a comparable relationship with Merged Drought Index (MDI) model from MTC, STC, and TC. However, individual indices SPI and SPTI are strongly associated with MTC and STC-based MDI compared to SPEI. Sen's slope shows the same trend across all approaches with minimal amplitude divergence. KGE was assessed using an average of one hundred thousand simulated values, and STC and TC demonstrated higher efficiency than MTC. But MTC has a lower error variance in contrast to STC and TC. Overall, the current study's findings validate that Merged Drought Index (MDI) based on MTC and proposed STC provides a better quantitative way to merge three separate drought indices into a single index. So, MDI successfully captured recorded drought episodes throughout the research locations, indicating that the merging method can be a workable option to identify drought accurately.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
In this study, we propose to introduce and apply an accurate numerical procedure to solve the mathematical model of fractional order, which describes the electrical circuits composed of resistors and ...inductors (RL) driven by a voltage of the current source. Our research is based on the spectral collocation method, which utilizes the advantageous characteristics of the third‐order Chebyshev polynomials. Some convergence analysis and error theorems are presented. The proposed research concludes by comparing the approximate solutions obtained for the considered model to the exact solution obtained in the classical case. The derived conclusions are further subjected to illustrative graphical and numerical examinations.
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In this article, we describe an online English collocation explorer developed to help English L2 learners produce correct and appropriate collocations. Our tool, which is able to visually represent ...relevant correct/incorrect collocations on a single webpage, was designed based on the notions of collocation clusters and intercollocability proposed by Cowie and Howarth. As they pointed out, in a collocation cluster L2 learners generally cannot distinguish true collocations (e.g., tell truth, state truth, and state fact) from impossible combinations (e.g., *say fact and *say truth). Accordingly, our tool applies natural language processing techniques to construct collocation clusters to enable learners to easily differentiate between correct and incorrect pairs. Relying on data from a reference corpus, our system instantaneously processes the collocability of users' target combination (verb-noun or adj-noun) and all other relevant words and presents true/false collocations that L2 learners should master/avoid. To assess our tool, we investigated its performance in detecting and correcting learners' V-N and A-N errors, with results comparable to those of most previous studies. Piloted using a sample of 13 intermediate- or upper-intermediate level English as a foreign language learners, our tool was found to help them self-correct their collocation errors effectively. Compared with similar tools or approaches, our tool requires much less data resources, but still demonstrates a remarkable capability to detect/correct errors and generate useful collocational knowledge in English.
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BFBNIB, NUK, PILJ, SAZU, UL, UM, UPUK
In this paper, a deep collocation method (DCM) for thin plate bending problems is proposed. This method takes advantage of computational graphs and backpropagation algorithms involved in deep ...learning. Besides, the proposed DCM is based on a feedforward deep neural network (DNN) and differs from most previous applications of deep learning for mechanical problems. First, batches of randomly distributed collocation points are initially generated inside the domain and along the boundaries. A loss function is built with the aim that the governing partial differential equations (PDEs) of Kirchhoff plate bending problems, and the boundary/initial conditions are minimised at those collocation points. A combination of optimizers is adopted in the backpropagation process to minimize the loss function so as to obtain the optimal hyperparameters. In Kirchhoff plate bending problems, the C1 continuity requirement poses significant difficulties in traditional mesh-based methods. This can be solved by the proposed DCM, which uses a deep neural network to approximate the continuous transversal deflection, and is proved to be suitable to the bending analysis of Kirchhoff plate of various geometries.
The paper is concerned with a new numerical method, NURBS based hybrid collocation–Galerkin method (NURBS-HCGM), to solve the in-plane motion problem of elastic solids. It combines the merits of the ...so-called scaled boundary finite-element method (SB-FEM) and the isogeometric collocation method. For the analysis, the boundary scaling technique of SB-FEM is adopted. It leads to a formulation, where only the boundary of a structure is discretized. Here, the NURBS basis functions are employed for the description of the geometry of the boundary as well as for the approximation of the displacements at the boundary. This is in accordance with the boundary representation modeling technique, which is commonly employed in computer aided design software. The inner domain is described by a radial scaling parameter. Applying the weak form only in circumferential direction the governing partial differential equations of elasticity are transformed to an ordinary differential equation (ODE) of Euler type, where the unknown displacements are a function of the radial scaling parameter. In the present work a NURBS based collocation scheme is introduced to solve this equation. NURBS basis functions are suggested for the approximation of the displacements in scaling direction. The higher continuity provided by NURBS allows to use collocation to solve the ODE directly instead of using the weak form in scaling direction. The proposed approach is validated by comparison with the eigenvalue solution of the ODE. It is remarked that the eigenvalue solution is restricted to linear problems, whereas the proposed method could be extended to nonlinear problems. In general, the presented formulation will allow to model patches bounded by an arbitrary number of contour boundaries. The accuracy of the proposed approach is analyzed and estimated with respect to analytical solutions. The computational cost is investigated with the help of numerical examples and is compared to isogeometric Galerkin approach.
•An analysis method for the boundary representation modeling technique is proposed.•It combines isogeometric analysis and the scaled boundary finite element method.•A surface oriented formulation to analyze 2D solids is presented.•A collocation approach is applied to treat the ODE for the inner domain.•The formulation allows to model patches with an arbitrary number of boundaries.
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The present paper presents a robust multi-patch formulation based on the isogeometric collocation (IGA-C) method for the solution of linear, spatial Timoshenko beam structures with complex geometry. ...The proposed approach is based on the combination of the (local) Bishop frame with the exponential map for SO(3) (Rodrigues formula) to compute the beam curvature and its derivative. This choice permits bypassing known issues related to the Serret–Frenet (SF) frame (in cases where the local beam curvature vanishes) and does not require the Darboux vector and its derivative, which are both affected by limitations of the SF frame. Moreover, in contrast to existing formulations mostly derived in the local SF frame, here the formulation is consistently derived by linearizing the governing equations in the material setting, where the multi-patch coupling can be enforced in a straightforward way. Numerical tests are performed on complex spatial curved beams, including a demanding biomechanical problem of a braided stent, proving the accuracy and the robustness of the formulation.
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Root zone soil moisture (RZSM) is a vital variable for vegetation growth, drought monitoring and agricultural water management. Satellite remote sensing measures soil moisture at the surface layer, ...while RZSM is derived usually by model-based simulations. Here, we provide the first comprehensive evaluation of eight RZSM products at a global scale, including GLDAS NOAH, ERA-5, MERRA-2, NCEP R1, NCEP R2, JRA-55, SMAP level 4 and SMOS level 4 datasets. An in-situ validation based on the stations from the International Soil Moisture Network (ISMN) and a triple collocation (TC) evaluation are both conducted to assess the accuracy of these RZSM products. SMAP exhibits the median highest correlation and the median lowest RMSE with in-situ stations over North America. In the TC analysis, MERRA-2 shows the highest median correlation and the median lowest error standard deviation with the unknown truth, followed by GLDAS, SMAP, JRA-55 and ERA-5. A temporal pattern analysis indicates that SMOS has a dry bias relative to other datasets and NCEP R1 has larger seasonal variations relative to other datasets over Asia and North America. The TC analysis indicates that MERRA-2, SMAP, GLDAS, JRA-55, and ERA-5 have better performance relative to other datasets. SMAP is not as good as GLDAS, MERRA-2 and JRA-55 in RZSM estimation over forest areas. The possible factors influencing RZSM performance are discussed, including precipitation forcing, assimilated observations, radio frequency interference issue and validation methods. These results and conclusions may provide new insights for the improvement of model-based RZSM estimation.
•Evaluation of eight RZSM products by ISMN stations and triple collocation globally.•SMAP has highest correlation and lowest RMSE with in-situ stations in North America.•MERRA-2 performs better than other products in TC, followed by GLDAS and SMAP.•Triple collocation and in-situ validation could complement each other.
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