CPU Time Cost Research Articles

Regularization techniques and inverse approaches in 3D Traction Force Microscopy

The conception of inverse methods in the context of Traction Force Microscopy (TFM) is influenced by multiple factors, such as nonlinear effects, dimensionality (2D/3D), and regularization/constraints, amongst others. Solving the inverse problem often requires the inversion of a matrix, and the presence of noise in the measured displacements can lead to unrealistic reconstructed tractions. To address this issue, Tikhonov regularization is commonly used, penalizing high norm values of computed tractions. The aim of this study is to compare the performance of different inverse methodologies (including some original variations) in 3D TFM, considering constraint imposition and regularization along the formulation. The different methodologies are numerically elaborated within a novel combined Newton–Raphson/Finite Element Method scheme that provides converged solutions in few iterations. The impact of constraint imposition and regularization in traction reconstruction is evaluated in terms of accuracy, efficiency (CPU time), and inherent characteristics of the methodology. Results show that, applying regularization and constraints (based on the fulfillment of fundamental principles) provides the best traction reconstruction, while simultaneously ensuring an optimum estimate of the maximum traction, at the cost of high CPU time and low efficiency. Moreover, regularization-based methods introduce the challenge of calibrating the regularization parameter, usually done under subjective criteria. On the other hand, non-regularized but constrained methods may represent a nice compromise between accuracy and efficiency, while avoiding pre-processing and calibration of such regularization parameter. It is emphasized the importance of considering not only traction reconstruction quality but also the efficiency and complexity of implementation (intrusiveness) when selecting an appropriate inverse method for TFM analysis.

An absolutely convergent fixed-point fast sweeping WENO method on triangular meshes for steady state of hyperbolic conservation laws

High order accuracy fast sweeping methods have been developed in the literature to efficiently solve steady state solutions of hyperbolic partial differential equations (PDEs) on rectangular meshes, while they were not available yet on unstructured meshes. In this paper, we extend high order accuracy fast sweeping methods to unstructured triangular meshes by applying fixed-point iterative sweeping techniques to a fifth-order finite volume unstructured WENO scheme, for solving steady state solutions of hyperbolic conservation laws. Similar as other fast sweeping methods, fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady state solutions. An advantage of fixed-point fast sweeping methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. This also provides certain convenience in designing high order fast sweeping methods on unstructured meshes. As in the first order fast sweeping methods on triangular meshes, we introduce multiple reference points to determine alternating sweeping directions on unstructured meshes. All the cells on the mesh are ordered according to their centroids' distances to those reference points, and the resulted orderings provide sweeping directions for iterations. To make the residue of the fast sweeping iterations converge to machine zero / round off errors, we follow the approach in our early work of developing the absolutely convergent fixed-point fast sweeping WENO methods on rectangular meshes, and adopt high order WENO scheme with unequal-sized sub-stencils, specifically here a fifth-order finite volume unstructured WENO scheme for spatial discretization. Extensive numerical experiments, including problems with complex domain geometries, are performed to show the accuracy, computational efficiency, and absolute convergence of the presented fifth-order fast sweeping scheme on triangular meshes. Furthermore, the proposed method is compared with the forward Euler time marching method and the popular third order total variation diminishing Rung-Kutta (TVD-RK3) time-marching method for steady state computations. Numerical examples show that the developed fixed-point fast sweeping WENO method is the most efficient scheme among them, and especially it can save up to 70% CPU time costs than TVD-RK3 in order to converge to steady state solutions.

Rotating Stall Inception Prediction Using an Eigenvalue-Based Global Instability Analysis Method

The accurate prediction of rotating stall inception is critical for determining the stable operating regime of a compressor. Among the two widely accepted pathways to stall, namely, modal and spike, the former is plausibly believed to originate from a global linear instability, and experiments have partially confirmed it. As for the latter, recent computational and experimental findings have shown it to exhibit itself as a rapidly amplified flow perturbation. However, rigorous analysis has yet to be performed to prove that this is due to global linear instability. In this work, an eigenanalysis approach is used to investigate the rotating stall inception of a transonic annular cascade. Steady analyses were performed to compute the performance characteristics at a given rotational speed. A numerical stall boundary was first estimated based on the residual convergence behavior of the steady solver. Eigenanalyses were then performed for flow solutions at a few near-stall points to determine their global linear stability. Once the relevant unstable modes were identified according to the signs of real parts of eigenvalues, they were examined in detail to understand the flow destabilizing mechanism. Furthermore, time-accurate unsteady simulations were performed to verify the obtained eigenvalues and eigenvectors. The eigenanalysis results reveal that at the rotating stall inception condition, multiple unstable modes appear almost simultaneously with a leading mode that grows most rapidly. In addition, it was found that the unstable modes are continuous in their nodal diameters, and are members of a particular family of modes typical of a dynamic system with cyclic symmetries. This is the first time such an interesting structure of the unstable modes is found numerically, which to some extent explains the rich and complex results constantly observed from experiments but have never been consistently explained. The verified eigenanalysis method can be used to predict the onset of a rotating stall with a CPU time cost orders of magnitude lower than time-accurate simulations, thus making compressor stall onset prediction based on the global linear instability approach feasible in engineering practice.

An adaptive support domain for the in-compressible fluid flow based on the localized radial basis function collocation method

In this paper, an adaptive support domain scheme is proposed to the in-compressible fluid flow based on the localized radial basis function collocation method (LRBFCM). The idea of the adaptive support domain avoids complex mathematical derivations of the finite difference method (FDM). Unlike other upwind schemes in the LRBFCM, the support domain uses the sampling nodes similar to different finite difference schemes, and the improved LRBFCM is further applied to the in-compressible fluid flows. Considering the Kronecker tensor product and Cholesky decomposition techniques, the improved LRBFCM can be easily used to the lid-driven problems with Reynolds numbers up to 100,000 by a simple mathematical derivation. The proposed LRBFCM has an adaptive support domain similar to the finite difference schemes with much less CPU time costs.

Research on linguistic multi-attribute decision making method for normal cloud similarity

Aiming at the uncertainty linguistic transformation problem in multi-attribute decision making, a decision-making method based on normal cloud similarity was proposed. Firstly, starting from the normal cloud characteristic curves, a normal cloud similarity measurement method based on Wasserstein distance is proposed by combing with the normal cloud entropy-containing expectation curve, which is using the Wasserstein distance to characterize the similarity characteristics of probability distribution. The properties of the proposed similarity measure are discussed in the paper. Secondly, the performance of the proposed method is compared and analyzed with the existed methods by numerical simulation experiment and time series data classification experiment. The experimental results show that the proposed method has good similarity discrimination ability, high classification accuracy and low CPU time cost. Finally, the method was successfully applied into linguistic multi-attribute decision making, and TOPSIS thought is used to compare and rank the schemes, so as to realize the final decision.

A fast procedure for the construction of quadrature formulas for bandlimited functions

We introduce an efficient scheme for the construction of quadrature rules for bandlimited functions. While the scheme is predominantly based on well-known facts about prolate spheroidal wave functions of order zero, it has the asymptotic CPU time estimate O(nlog⁡n) to construct an n-point quadrature rule. Moreover, the size of the “nlog⁡n” term in the CPU time estimate is small, so for all practical purposes the CPU time cost is proportional to n. The performance of the algorithm is illustrated by several numerical examples.

Weight splitting iteration methods to solve quadratic nonlinear matrix equation [formula omitted

The quadratic nonlinear matrix equationQ(Y)=MY2+NY+P,occurs in many applications such as the Quasi-Birth-Death processes, pseudo-spectra for quadratic eigenvalue problems and the quadratic eigenvalue problems. In this work, we propose efficient parametric iterative methods for finding the solution of this quadratic matrix equation based on weight splitting (WS) on matrices M and N, separately. We show that the proposed methods converge to the solution of the quadratic nonlinear matrix equation under conditions. Every iteration in proposed methods requires the solution of one or two linear matrix equations. Finally, various numerical examples indicate that the obtained methods in terms of CPU time, accuracy and computational cost are superior to the famous methods.

Fast Electromagnetic Validations of Large-Scale Digital Coding Metasurfaces Accelerated by Recurrence Rebuild and Retrieval Method

The recurrence rebuild and retrieval method (R3M) is proposed in this paper to accelerate the electromagnetic (EM) validations of large-scale digital coding metasurfaces (DCMs). R3M aims to accelerate the EM validations of DCMs with varied codebooks, which involves the analysis of a group of similar but not identical structures. The method transforms general DCMs to rigorously periodic arrays by replacing each coding unit with the macro unit, which comprises all possible coding states. The system matrix corresponding to the rigorously periodic array is globally shared for DCMs with arbitrary codebooks via implicit retrieval. The discrepancy of the interactions for edge and corner units are precluded by the basis extension of periodic boundaries. Moreover, the hierarchical pattern exploitation (HPE) algorithm is leveraged to efficiently assemble the system matrix for further acceleration. Due to the fully utilization of the rigid periodicity, the computational complexity of R3M-HPE is theoretically lower than that of $\mathcal{H}$-matrix within the same paradigm. Numerical results for two types of DCMs indicate that R3M-HPE is accurate in comparison with commercial software. Besides, R3M-HPE is also compatible with the preconditioning for efficient iterative solutions. The efficiency of R3M-HPE for DCMs outperforms the conventional fast algorithms in both the storage and CPU time cost.

Hierarchical Pattern Exploitation for Efficient Electromagnetic Analysis of Finite Periodic Array

The hierarchical pattern exploitation (HPE) method is proposed for the efficient electromagnetic analysis of finite periodic arrays. Instead of taking advantage of the Toeplitz symmetry by discretizing integral equations (IEs), HPE partitions the array hierarchically to form considerably larger geometrical repetitions, which result in identical interaction blocks inside the system matrix. These interaction blocks representing patterns are characterized and hashed to generate a directory, which maps the patterns to the references of the submatrices. Due to the large proportion of repetitions in hierarchical matrix ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}$ </tex-math></inline-formula> -matrix), HPE accelerates the matrix assembly and reduces the required storage drastically. Numerical examples show that the HPE outperforms the classical <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}$ </tex-math></inline-formula> -matrix by a large margin with much less storage and CPU time cost for the analysis of finite periodic arrays. Compared with the multilevel fast multipole algorithm (MLFMA) and the characteristic basis functions method (CBFM), HPE also demonstrates competitive performance, which validates the effectiveness and efficiency of the proposed method.

Modeling Graphene-Based THz Devices With an Improved Surface Boundary Condition in Leapfrog FDTD Scheme

Finite-difference time-domain (FDTD) method is an efficient full-wave numerical method for simulating graphene-based optoelectronic devices. The previous investigation on modeling graphene in leapfrog FDTD methods shows that dispersive errors increase with tuning up the chemical potential of graphene. In this article, a new implementation of graphene’s surface boundary condition (SBC) is proposed to reduce the dispersive errors in the leapfrog FDTD update scheme. To achieve lower dispersive errors, the graphene layer is positioned exactly on the grid lattice of transverse magnetic fields, and each of the transverse magnetic fields is split into two fields on the two sides of graphene. The split magnetic fields are updated using forward and backward differences in the electric fields on the normal direction of graphene plane. Numerical analysis shows that the maximum dispersive error of the proposed graphene model is reduced to only 37.5% of the other publicly reported graphene models in the leapfrog FDTD update scheme. However, the CPU time cost keeps almost the same as the other graphene models. The proposed implementation of the graphene SBC model also shows less CPU time consumption compared with the graphene SBC model which requires to solve a linear system and breaks the leapfrog FDTD update scheme, while the maximum numerical error is the same. The proposed graphene model is applied to simulate graphene terahertz (THz) surface plasmon resonances and modulators to demonstrate the applications in accurately modeling and analyzing graphene-based THz devices.

Efficient Boosted DC Algorithm for Nonconvex Image Restoration with Rician Noise

Image deblurring under Rician noise has attracted considerable attention in imaging science. Frequently appearing in medical imaging, Rician noise leads to an interesting nonconvex optimization problem, termed as the MAP-Rician model, which is based on the Maximum a Posteriori (MAP) estimation approach. As the MAP-Rician model is deeply rooted in Bayesian analysis, we want to understand its mathematical analysis carefully. Moreover, one needs to properly select a suitable algorithm for tackling this nonconvex problem to get the best performance. This paper investigates both issues. Indeed, we first present a theoretical result about the existence of a minimizer for the MAP-Rician model under mild conditions. Next, we aim to adopt an efficient boosted difference of convex functions algorithm (BDCA) to handle this challenging problem. Basically, BDCA combines the classical difference of convex functions algorithm (DCA) with a backtracking line search, which utilizes the point generated by DCA to define a search direction. In particular, we apply a smoothing scheme to handle the nonsmooth total variation (TV) regularization term in the discrete MAP-Rician model. Theoretically, using the Kurdyka--Lojasiewicz (KL) property, the convergence of the numerical algorithm can be guaranteed. We also prove that the sequence generated by the proposed algorithm converges to a stationary point with the objective function values decreasing monotonically. Numerical simulations are then reported to clearly illustrate that our BDCA approach outperforms some state-of-the-art methods for both medical and natural images in terms of image recovery capability and CPU-time cost.

A Fixed-Point Fast Sweeping WENO Method with Inverse Lax-Wendroff Boundary Treatment for Steady State of Hyperbolic Conservation Laws

Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations (PDEs). The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions. A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. Hence, they are easy to be applied to a general hyperbolic system. To deal with the difficulties associated with numerical boundary treatment when high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains, inverse Lax-Wendroff (ILW) procedures were developed as a very effective approach in the literature. In this paper, we combine a fifth-order fixed-point fast sweeping WENO method with an ILW procedure to solve steady-state solution of hyperbolic conservation laws on complex computing regions. Numerical experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the grids. Numerical results show high-order accuracy and good performance of the method. Furthermore, the method is compared with the popular third-order total variation diminishing Runge-Kutta (TVD-RK3) time-marching method for steady-state computations. Numerical examples show that for most of examples, the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.

Simulation of electromagnetic wave propagations in negative index materials by the localized RBF-collocation method

In this paper, a fast simulation of the electromagnetic wave propagation in negative index materials is developed by the localized meshfree radial basis function (RBF) collocation method. The perfect match layer is considered on the boundary in a unit cell to avoid the wave reflection. The experienced shape parameter is validated by simple test functions. The CPU time costs are compared with those of FEM. Numerical examples of different scatterers are presented to demonstrate the efficiency of the proposed method.

The ICSCREAM Methodology: Identification of Penalizing Configurations in Computer Experiments Using Screening and Metamodel—Applications in Thermal Hydraulics

In the framework of risk assessment in nuclear accident analysis, best-estimate computer codes associated with probabilistic modeling of uncertain input variables are used to estimate safety margins. Often, a first step in such uncertainty quantification studies is to identify the critical configurations (or penalizing, in the sense of a prescribed safety margin) of several input parameters (called scenario inputs) under the uncertainty of the other input parameters. However, the large CPU-time cost of most of the computer codes used in nuclear engineering, as the ones related to thermal-hydraulic accident scenario simulations, involves developing highly efficient strategies. This work focuses on machine learning algorithms by way of a metamodel-based approach (i.e., a mathematical model that is fitted on a small sample of simulations). To achieve it with a very large number of inputs, a specific and original methodology called Identification of penalizing Configurations using SCREening And Metamodel (ICSCREAM) is proposed. The screening of influential inputs is based on an advanced global sensitivity analysis tool (Hilbert-Schmidt Independence Criterion importance measures). A Gaussian process metamodel is then sequentially built and used to estimate within a Bayesian framework the conditional probabilities of exceeding a high-level threshold according to the scenario inputs. The efficiency of this methodology is illustrated with two high-dimensional (around a hundred inputs) thermal-hydraulic industrial cases simulating an accident of primary coolant loss in a pressurized water reactor. For both use cases, the study focuses on the peak cladding temperature (PCT), and critical configurations are defined by exceeding the 90%-quantile of the PCT. In both cases, using only around one thousand code simulations, the ICSCREAM methodology allows one to estimate the impact of the scenario inputs and their critical areas of values.

A depth-integrated non-hydrostatic model for nearshore wave modelling based on the discontinuous Galerkin method

In this study, a convergent two-dimensional depth-integrated non-hydrostatic shallow water model is discretized using the discontinuous Galerkin (DG) method. This model can account for the dispersive effects by including a non-hydrostatic pressure component and can be used for the simulation of weakly dispersive water waves. Additionally, this model is based on the fractional step method. In the first step, the traditional nonlinear shallow water model plus a transport equation for vertical momentum is discretized by the DG method, and the resulting semi-discrete system is evolved in time by the explicit third-order strong-stability-preserving Runge-Kutta (SSPRK3) method to obtain the provisional solution. In the second step, the provisional solution is corrected by satisfying a divergence constraint for the velocity. The latter step is conducted after application of the DG discretization to an elliptic equation regarding the non-hydrostatic pressure. Both theoretical and experimental tests are carried out to validate the proposed model. The results indicate that for smooth problems, our model can arrive at a given error tolerance by using a mesh with fewer degrees of freedom (DOFs) at the cost of a smaller CPU time when a higher approximation order is adopted and can properly simulate wave run-up, diffraction, refraction, and focusing.

CPU-time and RAM memory optimization for solving dynamic inverse problems using gradient-based approach

Numerical solution of inverse problem for 2D acoustic system of conservation laws by gradient type method requires storage of O(N3) elements which is crucial on large grids with O(N) points in single dimension. In this article we present an approach to save twice memory on the stage of adjoint problem and gradient calculation and compare it with usual approach in memory and CPU time cost. Numerical comparison for CPU time and memory of one step of iteration process which consists of direct problem solution, adjoint problem solution and calculation of the gradient are presented.

Second-order two-scale analysis for heating effect of periodical composite structure

The multi-scale analysis method is presented for the heat transfer problems of periodical composite structures under irradiation heating. This kind of physical problem can be described as a class of parabolic equations with small periodical oscillating coefficients equipped with the mixed boundary conditions of both Dirichlet and Neumann type. It needs more expensive cost in both computer's memory and CPU time to solve these problems by the usual finite difference method or finite element method, since it is necessary to partition the considering geometrical region finer to capture the heat conduction behavior at small scale. To reduce the computational complexity, an approximate solution is derived by the asymptotic expansion technique. The convergence of asymptotic solution to exact solution is technically proved as the micro-scale parameter tends to zero. Based on the asymptotic expansion formula, a second-order two-scale finite element method is proposed to fully solve this problem in practice. Numerical results show convergent behavior of the proposed method.

A coupled permeability thresholding with effective medium theory upscaling technique for highly heterogeneous reservoirs

Reservoir simulations are performed with fine grid geological models as primary input into the reservoir simulators. These fine grid geological models contain ten million to one hundred million cells while current serial reservoir simulators can only handle one hundred thousand to one million cells. This disparity in scales prohibits direct input of fine grid geological models into serial simulators. However, traditional upscaling techniques are unable to reproduce the fine-scale dynamic flow behavior at the coarse-scale and up-scaled permeability values are dependent on the applied boundary conditions. The problem is exacerbated if isolated flow units are present. These problems motivated the need to couple the permeability thresholding and effective medium theory (EMT) approaches to develop a practical and accurate quasi-global single-phase permeability upscaling technique that uses the reservoir simulator in the upscaling process. The workflow involves identifying and making the no-flow cells slightly permeable. This is followed by defining two models for each region to be scaled up, one of which includes the heterogeneous permeability field in the region and the other the upscaled value. Also, a fine-scale gridded ring is constructed around the target region. Then the upscaled value of the permeability is resolved by minimizing the difference between the well water production rates (WWPR) of the two runs. The technique is shown to mimic the channel fine-scale permeability distribution at the coarse-scale. It is also shown to reproduce the physics of flow for two-phase flow problems as evident in the well oil production rate, injection well pressure and pressure distribution profile. Sensitivity analyses on different scale-up factors, well locations and sequentially active wells show the superiority of the novel technique over the upscaling and downscaling with effective medium theory (UDEMT) and local upscaling techniques - albeit at the cost of more CPU time. The new method takes more CPU time than local upscaling method but the accurate results justifies the extra CPU time. Our workflow should be extended to three dimensions and fractured reservoirs as well as coupled with existing upscaling techniques for near wellbore and relative permeability upscaling.

Synthesis of Large Unequally Spaced Planar Arrays Utilizing Differential Evolution With New Encoding Mechanism and Cauchy Mutation

This article presents a differential evolution algorithm with a new encoding mechanism and Cauchy mutation (DE-NEM-CM) for optimizing large unequally spaced planar array layouts with the minimum element spacing constraint. In the new encoding mechanism, each individual represents a certain element position rather than an entire array layout used in traditional stochastic optimization algorithms. Such an encoding mechanism has the following advantages: 1) in each individual updating, the array pattern can be efficiently evaluated by only considering the radiation contribution variation from one element movement, which can greatly reduce the computational time; 2) it naturally facilitates the generated new array layout in population updating to meet the minimum element spacing constraint, and 3) each individual is searched always in 2-D space as the array size increases. These advantages enable it to be very suitable for synthesizing large arrays. Besides, DE serves as a search engine, and Cauchy mutation with chaotic mapping is proposed to enhance the local search while preserving the diversity of the population. A set of experiments for synthesizing different types of unequally spaced planar arrays in both narrow-and broadband applications are conducted. Synthesis results show that the proposed method achieves much lower sidelobe level than some state-of-the-art stochastic optimization methods for all the test cases. Importantly, the proposed method is much more efficient than conventional stochastic optimization algorithm especially for the case of synthesizing large unequally spaced planar array layouts. A array layout optimization with more than 1000 elements can be achieved within acceptable CPU time cost, which has not yet been reported for the existing stochastic optimization methods without resorting to supercomputing facilities.

Построение рецепторных геометрических моделей объектов сложных технических форм

In this paper the question related to the use of receptor (voxel) method for geometric modeling to solve practical design problems has been considered. The use of receptor methods is effective in solving a certain class of problems, primarily the problems of automated layout. The complexity of this method’s practical use is due to the fact, that receptor geometric models are never the primary ones. They are formed based on parametric models specified by designer. Receptor models are the internal machine ones. The main problem that prevents the widespread use of the receptor method is the lack of universal methods for converting parametric models into the receptor ones. Available publications show that in solving practical problems various authors have developed their own methods for creating receptor models for objects of "primitives" and "composition of primitives" classes. Therefore, it is extremely urgent to solve the problem of developing a universal method of forming receptor models for objects of complex technical forms. The essence of the proposed method is the transformation of a solid-state model created in a CAD system into a receptor matrix. First in the physical one, in which the solid-state model is discretized into cubes with receptor sizes, and then in the mathematical one — a three-dimensional array with binary codes of zeros and ones. The creation of a physical receptor matrix is carried out by means of the CAD-system itself, allowing diagnose the belonging of a single receptor to a simulated object. The fact of intersection or non-intersection a given position by a single receptor is encoded by "1" and "0" respectively, and this information is transferred to a mathematical receptor model (3-dimensional binary array). This calculation procedure is programmed in the form of a macro, providing a given position of a single receptor and fixing the fact of its intersection with the solid-state model. Have been demonstrated examples for described method’s practical application, and has been carried out CPU time cost estimation for the construction of a physical receptor model depending on the receptor size and object geometric complexity. Actions on data transformation from a solid-state model to a receptor one have been implemented in the form of C# programs.