Algorithm For Numerical Solution Research Articles

Radially transverse isotropic inclusions in isotropic elastic media: Local fields, neutral inclusions, effective elastic properties

Radially transverse isotropic inclusions in homogeneous isotropic elastic host media are considered. Mellin transform method is used for solution of the volume integral equation of the problem for an isolated inclusion subjected to a constant external stress (strain) field. The tensor structure of the solution is revealed with precision to three scalar functions of the radial coordinate, and the system of ordinary differential equations for these functions is derived. For multilayered radially transverse isotropic inclusions with constant elastic coefficients inside layers, explicit solution of these equations is obtained. An efficient numerical algorithm of solution for inclusions with an arbitrary number of the layers is proposed. Neutral inclusions that do not disturb homogeneous external fields applied to the medium are considered. It is shown that an inclusion with an isotropic core and radially transverse isotropic external layer can be weak neutral by appropriate choice of the layer elastic constants. The effective field method is used for determination of the effective elastic stiffness tensor of a homogeneous isotropic medium containing a random set of radially transverse isotropic inclusions.

Robust image segmentation and bias field correction model based on image structural prior constraint

In this paper, we propose an advanced variational model for image segmentation and bias correction. In contrast to the majority of existing level set segmentation models that only consider illumination bias fields, we additionally consider the impact of image reflectance on segmentation accuracy. Our method is capable of effectively segmenting images with blurry edge structures affected by non-uniform illumination. In order to enhance segmentation efficiency, we directly segment the underlying structures of the images, construct spatial prior and apply adaptive regularization constraints on the structural component. Therefore, in the process of segmentation, the proposed algorithm can accurately identify object boundaries without being affected by the environment. Besides, the GL operator is applied to enhance the robustness of the model against noise. Furthermore, we use the alternating direction method of multipliers and the operator splitting algorithm for numerical solution. The experimental results obtained from various sorts of images illustrate that our model outperforms many leading-edge level set models with regard to robustness, corrected results and accuracy.

A Numerical Expedition through the Mathematical Representation of Complex Braided Morphometry—A Case Study of Brahmaputra River in India

The present work explores the process of mathematical representation for the complex geometry of a wide alluvial river with high braiding intensities. It primarily focuses on an approach to developing a numerical solution algorithm for representing the complex channel geometry of the braided Brahmaputra River. Traditional elliptic PDEs with boundary-fitted coordinate transformation were deployed, converting the non-uniform physical plane into a transformed uniform orthogonal computational plane. This study was conducted for the river channel reach with upstream and downstream nodes at Pandu and Jogighopa (reach length ~100 km), respectively, within the Assam flood plain in India, with fourteen measured river cross-sections for the year of 1997. The geo-referenced image covering the river stretch in 1997 was delineated using a ArcGIS software 9.0 tool by digitizing the bank lines. Stream bed interpolation was conducted by interpolating bed elevation from a bathymetrical database onto code-generated mesh nodes. Discretization of the domain was performed through the developed computer code, and the bed-level matrix was generated by the IDW method as well as the MATLAB tool using the nearest neighborhood technique. A mathematical representation of a digital terrain model was thus developed. This generated model was employed as a geometrical data input to simulate secondary flow utilizing 2D depth-averaged equations with the flow dispersion stress tensor as an extra source component, coming from curvilinear flow patterns caused by severe river braiding. The developed model may further be useful in mathematically representing the geometrical complexities of braided rivers with a relatively realistic assessment of the various parameters involved if deployed with improved river modeling with morphometric evolution.

Parallel Algorithms for Numerical Solution of Spatially Three-Dimensional Diffusion-Convection Equations in Coastal Systems Based on Splitting Schemes

Introduction. To prevent the occurrence and mitigate the consequences of hazardous and catastrophic phenomena associated with sediment transport in natural systems, it is necessary to develop operational and scientifically justified forecasts, identify critical states at which the emergence of emergency situations is possible. For these purposes, it is necessary to create an accurate and efficient toolkit, including algorithms for numerical solution of a model problem that takes into account the specifics of natural systems. In this work, parallel algorithms for numerical solution of a spatially three-dimensional diffusion-convection problem of sediment are presented, which allow a significant reduction in computation time (by more than 4 times) compared to calculations conducted using a sequential algorithm.Materials and Methods. For the parallel solution of the spatially three-dimensional diffusion-convection problem, an implicit splitting scheme is constructed, in which the original continuous problem is replaced by a chain of two-dimensional and one-dimensional problems. The splitting schemes proposed in the work are physically justified and take into account the specifics of coastal marine systems, for which the influence of micro-turbulent diffusion and advective transport of substances are comparable, and the Peclet number does not exceed unity when approximating real problems. For the parallel numerical implementation, a method of decomposing the grid domain into two families of vertical planes parallel to the coordinate planes Oxz and Oyz, combined with the Seidel method for solving two-dimensional grid problems in horizontal planes and the tridiagonal matrix algorithm when solving one-dimensional three-point problems in the vertical direction, is used. Within the framework of the parallel computing software implementation, a parallel algorithm is presented that implements the diffusion-convection problem on a computing system using MPI technology.Results. A comparative analysis of parallel and sequential algorithms is obtained using a model problem.Discussion and Conclusions. The developed software allows its practical use for solving specific hydrophysical problems, including as part of a software complex.

An adaptive phase-field method for structural topology optimization

In this work, we develop an adaptive algorithm for the efficient numerical solution of the minimum compliance problem in topology optimization. The algorithm employs the phase field approximation and continuous density field. The adaptive procedure is driven by two residual type a posteriori error estimators, one for the state variable and the other for the first-order optimality condition of the objective functional. The adaptive algorithm is provably convergent in the sense that the sequence of numerical approximations generated by the adaptive algorithm contains a subsequence convergent to a solution of the continuous first-order optimality system. We provide several numerical simulations to show the distinct features of the algorithm.

Cluster modeling of the interaction of statyonary SH-waves with a system of curvilinear cracks in a half-space

A method is proposed for solving problems of mathematical physics for semi-infinite media containing systems of curvilinear cuts. Based on singular integral equations (SIE), a unified approach to solving the problem has been developed. Numerical implementation was carried out using parallelisation. A graph of the duration of the initial cluster hour is given for calculating an array of searched functions in the context of a parabolic form as a function of the number of processes for one variant of implementation. It is shown that the entire algorithm scales well and has an efficient number of processes. The expected result was obtained, because the structure of the calculation process in models based on SIE is mostly unchanged and built on well-defined procedures. It turned out that 150–200 processes are effective. An accuracy of 10–12 was achieved with the number of collocation points of the contour of each section N = 300, because the algorithm for numerical solution of the problem uses interpolation by Chebyshev polynomials in accordance with the fact that the unknown function has a key feature at the ends of the section, which causes a higher speed of convergence of the algorithm. The study of the question of the further increase in the accuracy of the result was not conducted. The corresponding dynamic boundary value problems for a restrained and force-free half-plane are studied. The influence of the curvature of defects, their interaction and proximity of the boundary on the magnitude and nature of the change in the dynamic stress intensity coefficients was studied. To check the reliability of the algorithm, two tests were carried out: removing three inhomogeneities at a distance of 106 of their length from each other and saturating the system by increasing the number of geometrically equal reflectors to 13–15. When removed, the characteristics of each reflector tend to the characteristics of a single one, and when saturated, they tend to the results of the corresponding periodic problem. The agreement between the results showed good reliability of the algorithm. The proposed method can be used to assess the influence of various mechanical or geometric factors on the strength of bodies with defects.

An explicit fourth-order accurate compact method for the Allen-Cahn equation

<abstract><p>In this paper, we propose an explicit spatially fourth-order accurate compact scheme for the Allen-Cahn equation in one-, two-, and three-dimensional spaces. The proposed method is based on the explicit Euler time integration scheme and fourth-order compact finite difference method. The proposed numerical solution algorithm is highly efficient and simple to implement because it is an explicit scheme. There is no need to solve implicitly a system of discrete equations as in the case of implicit numerical schemes. Furthermore, when we consider the temporally accurate numerical solutions, the time step restriction is not severe because the governing equation is a second-order parabolic partial differential equation. Computational tests are conducted to demonstrate the superior performance of the proposed spatially fourth-order accurate compact method for the Allen-Cahn equation.</p></abstract>

A deep learning method for the dynamics of classic and conservative Allen-Cahn equations based on fully-discrete operators

The Allen-Cahn equation is a well-known stiff semilinear parabolic equation used to describe the process of phase separation and transition in phase field modeling of multi-component physical systems, while the conservative Allen-Cahn equation is a modified version of the classic Allen-Cahn equation that can additionally conserve the mass. As neural networks and deep learning techniques have achieved significant successes in recent years in scientific and engineering applications, there has been growing interest in developing deep learning algorithms for numerical solutions of partial differential equations. In this paper, we propose a novel deep learning method for predicting the dynamics of the classic and conservative Allen-Cahn equations. Specifically, we design two special convolutional neural network models, one for each of the two equations, to learn the fully-discrete operators between two adjacent time steps. The loss functions of the two models are defined using the residual of the fully-discrete systems, which result from applying the central finite difference discretization in space and the Crank–Nicolson approximation in time. This approach enables us to train the models without requiring any ground-truth data. Moreover, we introduce an effective training strategy that automatically generates useful samples along the time evolution to facilitate training of the models. Finally, we conduct extensive experiments in two and three dimensions to demonstrate outstanding performance of our proposed method, including its dynamics prediction and generalization ability under different scenarios.

A continuum theory for stripe-shaped magnetic domains in thin films

Starting from a potential-based kinematical description of continuously distributed stripe-shaped domain walls, a mesoscopic potential is derived through an averaging procedure. The resulting mesoscopic theory is formulated in terms of a few fields with physically meaningful microscopic interpretation. Based on a phenomenological closure-domain extension, a constrained energy minimization principle is introduced. A monolithic finite-element based numerical solution algorithm is proposed. Based on a consistent linearization of the numerical algorithm, quadratic convergence is obtained for the nonlinear solution procedure. A first simple examination of the theory is undertaken by simulating a thin film with spatially and temporally non-constant effective anisotropy. The results are found to be consistent with analytical predictions.

Solution of nonlinear fractional-order models of nuclear reactor with parallel computing: Implementation on GPU platform

This paper present novel parallel computing algorithms for numerical solution of nonlinear fractional-order (FO) models of nuclear reactor. These FO models arise by virtue of considering neutron transport in nuclear reactor as subdiffusion. The numerical computation of these FO models is carried out by parallelizing the sequential steps in Adams–Bashforth–Moulton method. A comparison of parallel execution on MATLAB and CUDA platforms is also given. The parallel computation is executed by harnessing multicore architecture of the general purpose graphics processing unit (GPGPU). A detailed analysis of the proposed procedure is presented considering different values of fractional derivative order in the fractional differential equations (FDEs). It is shown with extensive computational exercise that the proposed methodology achieves substantial speed up for all the three FDEs models used in nuclear reactor.

Variational iteration algorithm for numerical solutions of sixth and seventh order boundary value problems using shifted Vieta-Lucas polynomials

This study employs Shifted Vieta-Lucas Polynomials using the variational iteration approach to numerically resolve sixth and seventh order Boundary Value Problems (BVPs), The proposed method in the study is used, with the trial functions for the approximation being the shifted Vieta Lucas polynomials generated for the given boundary value problem. The study provides an efficient and accurate solutions to higher-order BVPs using a combination of Shifted Vieta-Lucas Polynomials and the variational iteration approach and finds applications in mathematics and other fields of sciences and engineering.

Tidal barrage operation optimization using moment-based control


 As decarbonization of energy systems becomes an imperative and the deployment of intermittent renewables increases, the operation of electrical grids becomes challenging. In this sense, tidal barrage schemes can supply clean and predictable energy with more flexibility than the more traditional wind and solar plants. However, the high infrastructure costs associated with tidal barrage plants hinders its deployment. An optimal operation is key to maximize energy output and assure the project’s feasibility.
 Recently, Ringwood and Faedo [3] introduced a novel approach to tidal barrage control by applying wave energy control techniques, with promising results. The model consists on a two-way tidal barrage scheme, where an analogy is made between latching/declutching algorithms applied in WEC control and holding/sluicing from tidal barrage. The optimization was done using the moment-based framework developed in [1] and [2], which can formulate linear and non-linear problems in a concave quadratic environment, thus allowing the application of efficient numerical solution algorithms. This preliminary study has a number of simplifying assumptions, leaving a pathway to further examine the optimal control problem.
 This paper extends the results of the analysis from [3] by adding complexities to the model. Crucially, the basin is modeled such as the area is a polynomial function of the water level, instead of assuming constant surface area. This will affect the operational head of the optimal solution, as the volume of water at the top of the basin will be higher than at the bottom. Another added constraint will be the minimum head that the turbine requires to generate electricity. Furthermore, a cost function is introduced to penalize the energy consumption of the sluice gates, as shown in figures 1.a) and 1.b). This not only increases the accuracy of the model, but in many cases improves the convergence of the algorithm by adding a quadratic term. Figure 2 shows the value of the overall energy yield (as a fraction of the energy output without penalty) and simulation run time (as a fraction of the run time without penalty) with different penalty coefficients.
 1.a) Sluice gate area without considering cost function
 1.b) Sluice gate area with penalty factor of 3
 
 
 2. Energy and run time variations with penalty
 
 

Robust optimal control of nonlinear fractional systems

Parameter uncertainty is inevitably found in almost every system and, if is ignored, it could jeopardize the effectiveness of control method. Motivated by this issue, we consider a robust optimal control problem governed by a nonlinear fractional system with uncertain parameters, where the system sensitivity with respect to the uncertain parameters is explicitly included in the cost functional. For this problem, we first prove that the system sensitivity can be expressed as the solution of an auxiliary fractional system. Then, we propose a numerical scheme for discretizing both the original and auxiliary fractional systems, resulting in a finite-dimensional optimization problem. Furthermore, a numerical solution algorithm based on gradients of the cost functional is developed for the resulting optimization problem. Finally, numerical results for two example problems are given to demonstrate the validity and applicability of the proposed algorithm.

Modeling and Simulation of Dynamics in Soft Robotics: a Review of Numerical Approaches

Abstract Purpose of review In this review, we briefly summarize the numerical methods commonly used for the nonlinear dynamic analysis of soft robotic systems. The underlying mechanical principles as well as the geometrical treatment tailored for soft robots are introduced with particular emphasis on one-dimensional models. Additionally, the review encompasses three-dimensional frameworks, available simulation packages, and various types of interaction models, shedding light on the design, actuation, motion control, and internal and external forces of soft robots. Recent findings Reduced-order models can offer high efficiency in characterizing nonlinear deformations, allowing convenient tailoring based on specific structural and material configurations. For pursuing high simulation accuracy and detailed mechanics, the finite element method proves to be a valuable tool through numerous off-the-shelf platforms. Furthermore, machine learning has emerged as a promising tool to effectively address the challenges within the mechanics community. Summary A wide range of kinematic and dynamic numerical models is available for simulating the behaviors of soft robots, offering exceptional adaptability to different geometries and structures based on existing modeling theories and numerical solution algorithms. However, the trade-off between computational complexity and simulation accuracy remains a challenge in achieving fast, accurate, and robust control of soft robots in complex environments.

CFD Simulation of Low Speed Turbulent Flow Problems Using Unsteady Rans and Large Eddy Simulation Approach

The present paper focuses on the recent development of an implicit pressure-based finite volume algorithm for numerical solution of Navier Stokes equation in an inertial frame of reference for prediction of unsteady incompressible flow problems. The algorithm uses boundary-conforming, multiblock structured grid with moving boundaries, collocated variable arrangement with momentum equations resolved along cartesian directions, second order accurate spatial and temporal discretisation schemes for the convective fluxes and a pressurevelocity solution strategy. Effect of turbulence is simulated using one of the two different approaches. In the Unsteady Reynolds Averaged (URANS) approach coupled to appropriate eddy viscosity based turbulence models, the Navier Stokes (NS) equations, averaged over the whole range of turbulent length scales of the flow, are solved numerically. On the other hand, in the Large Eddy Simulation (LES) approach, the model filtered 3D NS equations are directly solved for the flow variables to resolve the large scale turbulent motions whereas the transport processes at the fine subgrid scale level only are simulated using simple algebraic turbulence model. The capabilities and limitations of both the cost-effective URANS approach and the relatively expensive but rich in physics LES approach have been demonstrated for few application problems of engineering interest.

A nonlinear substructure method based on a computational plasticity framework for efficient analysis of structural assemblies

Substructure analysis reduces the computational order of a discretized structural domain from the full set of degrees of freedom needed to solve a boundary value problem (e.g., the displacements of all nodes in an FEA mesh) to a predefined and much smaller set of retained degrees of freedom. Given only one initial analysis considering all degrees of freedom, this technique reduces the computational cost associated with subsequent analyses of the same domain by eliminating degrees of freedom, usually internal to the domain, which are not essential for interfacing the domain with a larger system/assembly. For large multiscale applications, such as aircraft and automotive assemblies, substructure analysis enables efficient computation of both static and dynamic responses of larger assemblies consisting of one or more of these previously analyzed and dimensionally reduced domains. However, while existing methods are exact for linear problems (e.g., small-deformation linear elasticity), they are insufficient when considering large deformations or material nonlinearities. In this work, we develop a new nonlinear substructure method to consider general nonlinear responses by leveraging the mathematical framework developed for computational plasticity, including a decomposition of deformations, criteria for nonlinearity initiation, and evolution equations. While computational plasticity provides nonlinear constitutive relationships between six independent stress and strain components, we show that the same mathematical formulation can capture similar relations between an arbitrary number of forces and displacements (i.e., the retained degrees of freedom), and thus, can be implemented with the same numerical solution algorithms. As a notional example to emphasize both the generality of the aforementioned method and the application to design frameworks, we model an infilled lattice structure exhibiting local plasticity and internal nonlinear geometric effects.

Simulation of reaction initiation in powder compacting from the surface with composite formation in equivalent reaction cell

In the present work, we propose a new variant of the model of the composite synthesis under surface heating. We believe that the formation of the composition occurs at the level of reaction cell. The diffusion-controlled process of reduction of one material by another from an oxide is described in the framework of the problem with moving boundaries. It is assumed that the formation of the matrix composition is carried out by the diffusion mechanism. From the position of interfaces, we find the relative volume fractions of oxide phases and the fraction of volume occupied by the matrix. The averaging method of the analysis results at the reaction cell level makes it possible to use these data at the macroscopic level. Volume fractions of phases and average matrix composition (obtained by averaging over the area occupied by the matrix) provide values reflecting the composition of the composite at the macro level. The problem is solved numerically in dimensionless formulation. Dimensionless complexes of physical quantities are distinguished. The estimation of these parameters is performed. The conditions of correctness of the proposed approach are established from comparison of temporal and spatial scales of thermal and diffusion phenomena. A numerical algorithm for the joint solution of macro- and mesolevel problems has been developed. The proposed algorithm makes possible the investigation of the dynamics of composition changes at all points. The model is supplemented by the calculation of stresses and strains from the data on composition and temperature changes in reaction cells. Averaged values of stresses are transferred to macro level.

What Are the Balanced and Unbalanced Dynamics of a Tropical Cyclone?

Abstract Recently, Ji and Qiao took into account the unbalanced components and derived an extended Sawyer–Eliassen (SE) equation. This study developed a new derivation of this extended SE equation from the perspective of restoring forces, and gives a physical interpretation for the coefficients that appear in the SE equation. For an unbalanced vortex, we demonstrated that the thermodynamic fields are only determined by the distribution of gradient wind, and thus, the gradient wind and thermodynamic fields always remain in balance as the unbalanced vortex evolves. Consequently, we attributed the gradient wind imbalance to the agradient wind rather than to the thermodynamic fields. Subsequently, we explored the effect of the agradient wind on the secondary circulation, and showed that the agradient wind strengthens the secondary circulation in its vicinity, which can be explained as a consequence of the restoring forces and mass continuity. Furthermore, we speculated that the SE equation, together with the radial velocity equation, could reproduce the primary characteristic of the axisymmetric boundary layer dynamics by prescribing the parameterization of subgrid-scale turbulent mixing. Specifically, the noU_BL and noVa_BL experiments conducted by Fei et al. in an article published in 2021 were reinterpreted, and the oscillation wavelength of the agradient wind in the eyewall was approximated based on this framework. Additionally, a new numerical solution algorithm to overcome the hyperbolicity near the boundary layer was proposed. This study attempts to develop a complete dynamic theory for tropical cyclones in both qualitative and quantitative perspectives.

Gaussian Belief Space Path Planning for Minimum Sensing Navigation

In this article, we propose a path planning methodology for a mobile robot navigating through an obstacle-filled environment to generate a reference path that is traceable with moderate sensing efforts. The desired reference path is characterized as the shortest path in an obstacle-filled Gaussian belief manifold equipped with a novel information-geometric distance function. The distance function we introduce is shown to be an asymmetric quasi-pseudometric and can be interpreted as the minimum information gain required to steer the Gaussian belief. An RRT*-based numerical solution algorithm is presented to solve the formulated shortest-path problem. To gain insight into the asymptotic optimality of the proposed algorithm, we show that the considered path length function is continuous with respect to the topology of total variation. Simulation results demonstrate that the proposed method is effective in various robot navigation scenarios to reduce sensing costs, such as the required frequency of sensor measurements and the number of sensors that must be operated simultaneously.

Integrated Energy System Design of Low-carbon City Based on Multi-energy Complementary of Cooling Heating and Power

The model of urban integrated energy dispatching system is a mathematical model used to describe the distribution of energy flow in the system, which has the characteristics of high accuracy, strong nonlinearity, and non-convexity. However, in the optimal power flow and other optimization problems, the traditional numerical solution algorithm cannot be directly applied. The strong nonlinearity and non-convexity of the system power flow model bring great difficulties to the solution of the optimization problem. Based on the input and output theory of cooling, heating, and power complementary energy, this paper studies the coupling energy relationship and power flow of various energy devices in an urban multi-source microgrid. It establishes the reliability index of the system energy supply, starting from the integrated energy system structure of low-carbon cities. A bi-level optimization model considering the constraints of energy supply reliability is constructed, which provides a basis for the subsequent establishment of the optimal allocation model.