Number Of Unknowns Research Articles

On a space-time implementation of the wave equation using virtual elements

AbstractThe virtual element method (VEM) was developed not too long ago, starting with the paper (Beirão-da-Veiga et al. in SIAM J Numer Anal 51:794–812, 2013) related to elasticity in solid mechanics. The virtual element method allows to revisit the construction of different elements, however has so far not applied to space-time formulations for one-dimensional structural elements like strings, trusses and beams. Here we study several VEM elements suitable for space-time formulations that are build upon the Hamilton’s principle. It will be shown that these elements can be easily incorporated in classical finite element codes since they have the same number of unknowns. Furthermore, we show that the property of VEM to deal with non-conforming meshes is of special interest for holistic space time formulation: VEM formulations allow locally varying time discretizations (time increments) in a natural and efficient way.

Semi-analytical solution of Brent equations

A parametrization of Brent equations is proposed which admit for a several times reduction of the number of unknowns and equations. The arising equations are solved numerically, and for the resulting fast matrix multiplication algorithms many known values of rank are reproduced and even improved, in particular, the designs (4,4,4;48) and (2,4,5;32) are found.

Выбор критериев для сокращения неизвестных в методе Ньютона–Рафсона в задаче нахождения равновесного состава железной руды и топлива

Finding the equilibrium composition of a multicomponent system is required in various industries associated with chemical reactions. Reduction of the unknowns in the original nonlinear equation is possible if most of the unknowns receive values equal to zero as a result of the solution. This situation is typical to solve the problem of finding an equilibrium composition of iron ore and fuel, where the unknowns are the number of moles of all possible reaction products from given simple substances. The dependence of algorithmic complexity on the number of unknowns to solve a SLAEs is expressed through the function O(N3). Therefore, reduction of the unknowns will speed up the computational process. Thus, the solution to the problem of selecting criteria to reduce the number of the unknowns at each iteration of the Newton-Raphson method as applied to the problem of finding the equilibrium composition of iron ore and fuel is relevant. The matrix dimensionality reduction has been achieved using four strategies. The first three strategies involve finding a minimum threshold (removing values that most quickly tend to zero), after which the number of unknowns will decrease. The fourth strategy is associated with reducing the dimension of the original matrix. The results of numerical experiments to reduce the unknowns and speed up calculations are presented. Criteria have been found under which the calculation speed increases by 2–4 times and variables whose value is not equal to zero are not reduced. The criteria to reduce the unknowns implemented in the T-Energу software package make it possible to solve the problem of finding the equilibrium composition of iron ore and fuel 2–4,2 times faster. In this case, the obtained solution for the components of the equilibrium composition with the reduction of the unknowns corresponds to the complete solution with an accuracy of 10–3. Due to acceleration at the same time, it is possible to construct equilibrium compositions over a temperature range with a temperature step that is 4 times smaller.

Vibration analysis using the theory of exponential shear deformation for laminated plates

In this research, the free vibration response of laminated composite plates is studied using a refined exponential shear strain theory. The most interesting feature of this theory is that it allows exponential distributions of transverse shear strains, and verifies zero-shear boundary conditions on the plate surfaces without the use of shear correction factors. Some stiffness constants are written as a series. The number of independent unknowns in the present theory is four, compared with five in other shear deformation theories. The equations of motion are obtained from Hamilton's principle and Navier's method is used to determine the exact solution for antisymmetric cross-laminated plates. The numerical results found in the present analysis for free vibration are presented and compared with those available in the literature. The proposed theory is not only accurate, but also effective in predicting the fundamental frequencies of laminated composite plates.

COMPARATOR REGRESSION ANALYSIS IN CONDITIONS OF FUZZY INITIAL DATA

Abstract. The canonical task of regression analysis consists in studying the problem of estimation and mathematical description of the influence caused by one or more independent variables (predictors, explanatory variables) on the dependent (explained) variable. The subject of comparator identification problem is a specific modification of the standard regression analysis problem in the case when, for some reason, it is not possible to measure values of the dependent, explained variable. However, the problem of reconstructing a regression polynomial can be solved if it is possible to qualitatively compare the results of experiments and rank them, for example, in the order of increasing. Taking into account the results of ranking the values of the explained variable in different experiments, a corresponding system of inequalities is formed. The purpose of the research consists in developing a method for mathematical processing of the resulting system of inequalities for analytical assessment of the level of influence of explanatory variables on the explained variable. The method for solving the problem is based on transforming a system of inequalities into a system of linear algebraic equations. A significant drawback of the known approach to solving this problem consists in the fact that the resulting system of linear algebraic equations is an underdetermined one (the number of unknowns in this system exceeds the number of equations). At the same time, the system has an infinite number of solutions, among which there may be an uncontrollable number of unacceptable ones. The exhaustive search method for finding acceptable solutions is futile. In this regard, the problem of developing a correct method of comparator identification remains relevant. Another difficulty that accompanies the actual procedure of comparator identification when solving practical problems consists in the fact that measurements of values of the controlled parameters are not accurate. The uncertainty that arises in the conditions of a small sample of initial data can be removed using the tools of fuzzy mathematics. It is assumed that, based on preliminary research, a mathematical description of the corresponding membership function can be obtained for each of the controlled parameters. As a result, effective approaches to solving emerging problems have been proposed, based on the use of developed optimization procedures in the conditions of a small sample of fuzzy initial data.

3D Controlled‐source electromagnetic modelling in anisotropic media using secondary potentials and a cascadic multigrid solver

AbstractQuantitative interpretation of the data from controlled‐source electromagnetic methods, whether via forward modelling or inversion, requires solving a considerable number of forward problems, and multigrid methods are often employed to accelerate the solving process. In this study, a new extrapolation cascadic multigrid method is employed to solve the large sparse complex linear system arising from the finite element approximation of Maxwell's equations using secondary potentials. The equations using secondary potentials are discretized by the classic nodal finite element method on nonuniform rectilinear grids. The resulting linear systems are solved by the extrapolation cascadic multigrid method with a new prolongation operator and preconditioned Stabilized bi‐conjugate gradient method smoother. High‐order interpolation and global extrapolation formulas are utilized to construct the multigrid prolongation operator. The extrapolation cascadic multigrid method with the new prolongation operator is easier to implement and more flexible in application than the original one. Finally, several synthetic examples including layered models, models with anisotropic anomalous bodies or layers, are used to validate the accuracy and efficiency of the proposed method. Numerical results show that the extrapolation cascadic multigrid method improves the efficiency of 3D controlled‐source electromagnetic forward modelling a lot, compared with traditional iterative solvers and some state‐of‐the‐art methods or software (e.g., preconditioned flexible generalized minimal residual method, emg3d) in the considered models and grid settings. The efficiency benefit is more evident as the number of unknowns increases, and the proposed method is more efficient at low frequencies. The extrapolation cascadic multigrid method can also be used to solve systems of equations arising from related applications, such as induction logging, airborne electromagnetic, etc.

A new system of sylvester-like matrix equations with arbitrary number of equations and unknowns over the quaternion algebra

In this paper, we derive some practical necessary and sufficient conditions for the existence of a solution to a new system of coupled two-sided Sylvester-like matrix equations with arbitrary number of equations and unknowns over the quaternion algebra. As applications, we give some practical necessary and sufficient conditions for the existence of an η-Hermitian solution to a system of quaternion matrix equations in terms of ranks. We also use graphs to represent linear mappings associated with some Sylvester-type systems. As a special case of the main theorem, we prove a conjecture is correct, which was proposed in [Linear Algebra Appl. 2016;496:549–593]. The main findings of this paper widely extend almost all the known results in the literature.

Static and free vibration response of FGM plates using higher order shear deformation theory and strain-based finite element formulation

The primary objective and novelty of this work lie in the development of a new four-node quadrilateral finite element based on strain-based high-order shear deformation theory (HSDT). This is the first study to apply this innovative approach to analyze both the static and free vibration behaviors of functionally graded (FG) plates. Another key novelty is the reduction in the number of unknowns to five, unlike other high-order shear deformation theories that employ a larger number of unknowns. This reduction is achieved by applying the condition of zero transverse shear stress at the top and bottom free surfaces of the FG plate and by assuming that the transverse shear strains are sinusoidally distributed through the thickness. The material properties of FG plates are modeled to vary according to a simple power law based on the volume fractions of their constituents. The developed finite element possesses five degrees of freedom (DOFs) per node, resulting from the combination of two strain-based elements: the first is a membrane with two DOFs, and the second is a bending plate with three DOFs. The displacement fields of the proposed element are expressed using high-order terms based on the strain approach, which satisfy compatibility equations and rigid body modes. Furthermore, the concept of the neutral surface is introduced to eliminate membrane-bending coupling. The elementary stiffness and mass matrices are derived using both the total potential energy principle and Hamilton’s principle. The performance and convergence of the proposed element are validated through examples from the literature.

Using the Constrained Disorder Principle to Navigate Uncertainties in Biology and Medicine: Refining Fuzzy Algorithms.

Uncertainty in biology refers to situations in which information is imperfect or unknown. Variability, on the other hand, is measured by the frequency distribution of observed data. Biological variability adds to the uncertainty. The Constrained Disorder Principle (CDP) defines all systems in the universe by their inherent variability. According to the CDP, systems exhibit a degree of variability necessary for their proper function, allowing them to adapt to changes in their environments. Per the CDP, while variability differs from uncertainty, it can be viewed as a regulated mechanism for efficient functionality rather than uncertainty. This paper explores the various aspects of un-certainties in biology. It focuses on using CDP-based platforms for refining fuzzy algorithms to address some of the challenges associated with biological and medical uncertainties. Developing a fuzzy decision tree that considers the natural variability of systems can help minimize uncertainty. This method can reveal previously unidentified classes, reduce the number of unknowns, improve the accuracy of modeling results, and generate algorithm outputs that are more biologically and clinically relevant.

Способи зниження складності обчислень при розрахунках потокорозподілу в мережах нестисливої рідини при фіксованих коефіцієнтах гідравлічного опору

Based on the analysis of the process of solving the problem of flow distribution calcula-tion in compressible fluid systems, which boils down to finding the root of a nonlinear system of equations using the Newton — Raphson method, methods of reducing the complexity of calculations at the stages of the iterative process are proposed. The studies belong to the class of hydraulic systems for which the total pressure change on the branch elements is a monotonic smooth function of the flow rate G. For the iterations of the Newton-Raphson method, the stages are analyzed: linearization of the nonlinear system of equations, determination of the discrepancy and error estimation, determination of cost growth and obtaining a new iterative cost value. A method and algorithm for determining the discrepancy and estimating the error with linear complexity relative to the number of branches of graph E is proposed. A method of reducing the number of unknowns in the system of flow distribution equations at the stage of forming a system of linear equations based on the use of the nodal convolution method is described. A number of variants of heuristic algorithms for determining the parameter that takes into account the rate of in-crease in costs when forming a new iterative value of costs and their analysis are present-ed.

Three-dimensional airborne electromagnetic forward modeling based on multiscale hexahedral finite-element method

Slow forward modeling is the main factor that restricts the practical use of three-dimensional (3D) inversion and interpretation of airborne electromagnetic (AEM) data. To improve the modeling efficiency in 3D AEM, we propose a new multiscale finite-element (MsFE) method based on unstructured hexahedral meshes. Compared with the traditional 3D AEM forward modeling, the main advantage of our newly developed method is that it can simulate complex underground structures in the earth quickly. Since we can fit the earth's topography or the anomalous bodies underground using a small number of hexahedral grids, we can quickly model them using MsFE. The main idea of the MsFE forward modeling method is to construct an interpolation operator between a coarse and a dense mesh and use the interpolation operator to map the conventional FE coefficient matrix to the MsFE coefficient matrix and thus reduce the number of unknowns in the modeling process. This will vastly reduce the scale of the linear equations system. We validate our method by simulating a typical mountain peak model and demonstrate its effectiveness by simulating numerous synthetic models and a model from Voisey Bay's Ovoid sulfide deposit, Canada.

A fourth-order compact finite volume method on unstructured grids for simulation of two-dimensional incompressible flow

This paper introduces a novel and efficient compact reconstruction procedure for high-order finite volume methods applied to unstructured grids. In this procedure, we establish a set of constitutive relations that ensure the continuity of the reconstruction polynomial and its normal derivatives between adjacent elements at control points. The paper delves into the details of the fourth-order compact reconstruction method specifically designed for two-dimensional triangular grids. This method can be considered an extension of the one-dimensional compact scheme and cubic spline interpolation methods to two-dimensional triangular unstructured grids. In the cubic polynomial reconstruction, a two-dimensional cubic polynomial is reconstructed using the cell averages of elements in the standard stencil and the function values at control points. The determination of function values at control points is achieved by adhering to the constraint of normal derivative continuity. This reconstruction is solved using a relaxation iteration method and has been verified to be solvable for triangular grids. Compared to other fourth-order implicit compact polynomial reconstructions, our method requires solving a smaller number of unknowns, which means less computational cost in reconstruction. By applying this method to solve the Poisson equation, the linear convection equation, and incompressible flow benchmarks, it demonstrates that the proposed method exhibits the expected high-order accuracy and performs well in incompressible flow problems.

Development of an untargeted DNA adductomics method by ultra-high performance liquid chromatography coupled to high-resolution mass spectrometry

Genotoxicants originating from inflammation, diet, and environment can covalently modify DNA, possibly initiating the process of carcinogenesis. DNA adducts have been known for long, but the old methods allowed to target only a few known DNA adducts at a time, not providing a global picture of the “DNA adductome”. DNA adductomics is a new research field, aiming to screen for unknown DNA adducts by high resolution mass spectrometry (HRMS). However, DNA adductomics presents several analytical challenges such as the need for high sensitivity and for the development of effective screening approaches to identify novel DNA adducts. In this work, a sensitive untargeted DNA adductomics method was developed by using ultra-high performance liquid chromatography (UHPLC) coupled via an ESI source to a quadrupole-time of flight mass spectrometric instrumentation. Mobile phases with ammonium bicarbonate gave the best signal enhancement. The MS capillary voltage, cone voltage, and detector voltage had most effect on the response of the DNA adducts. A low adsorption vial was selected for reducing analyte loss. Hybrid surface-coated analytical columns were tested for reducing adsorption of the DNA adducts. The optimized method was applied to analyse DNA adducts in calf thymus, cat colon, and human colon DNA by performing a MSE acquisition (all-ion fragmentation acquisition) and screening for the loss of deoxyribose and the nucleobase fragment ions. Fifty-four DNA adducts were tentatively identified, hereof 38 never reported before. This is the first untargeted DNA adductomics study on human colon tissue, and one of the few untargeted DNA adductomics studies in the literature reporting the identification of such a high number of unknowns. This demonstrates promising results for the application of this sensitive method in future human studies for investigating novel potential cancer-causing factors.

Impact of waning immunity against SARS-CoV-2 severity exacerbated by vaccine hesitancy.

The SARS-CoV-2 pandemic has generated a considerable number of infections and associated morbidity and mortality across the world. Recovery from these infections, combined with the onset of large-scale vaccination, have led to rapidly-changing population-level immunological landscapes. In turn, these complexities have highlighted a number of important unknowns related to the breadth and strength of immunity following recovery or vaccination. Using simple mathematical models, we investigate the medium-term impacts of waning immunity against severe disease on immuno-epidemiological dynamics. We find that uncertainties in the duration of severity-blocking immunity (imparted by either infection or vaccination) can lead to a large range of medium-term population-level outcomes (i.e. infection characteristics and immune landscapes). Furthermore, we show that epidemiological dynamics are sensitive to the strength and duration of underlying host immune responses; this implies that determining infection levels from hospitalizations requires accurate estimates of these immune parameters. More durable vaccines both reduce these uncertainties and alleviate the burden of SARS-CoV-2 in pessimistic outcomes. However, heterogeneity in vaccine uptake drastically changes immune landscapes toward larger fractions of individuals with waned severity-blocking immunity. In particular, if hesitancy is substantial, more robust vaccines have almost no effects on population-level immuno-epidemiology, even if vaccination rates are compensatorily high among vaccine-adopters. This pessimistic scenario for vaccination heterogeneity arises because those few individuals that are vaccine-adopters are so readily re-vaccinated that the duration of vaccinal immunity has no appreciable consequences on their immune status. Furthermore, we find that this effect is heightened if vaccine-hesitants have increased transmissibility (e.g. due to riskier behavior). Overall, our results illustrate the necessity to characterize both transmission-blocking and severity-blocking immune time scales. Our findings also underline the importance of developing robust next-generation vaccines with equitable mass vaccine deployment.

An Efficient Convergent Approach for Difference Delayed Reaction-Diffusion Equations

It is usually not possible to solve partial differential equations, especially the delay type, with analytical methods. Therefore, in this article, we present an efficient method for solving differential equations of the difference delayed reaction-diffusion type, which can be generalized to other delayed partial differential equations. In the proposed approach, we first convert the delayed equation into an equivalent non-delayed equation by inserting the corresponding delay function with an effective technique. Then, using a pseudo-spectral method, we discretize the obtained equation in the Legendre-Gauss-Lobatto collocation points and present an algebraic system with an equal number of equations and unknowns which can be solved by quasi-Newton methods such as Levenderg-Marquardt algorithm. The approximate solutions can be obtained with exponential accuracy. The convergence analysis of the method is fully discussed and four examples are presented to evaluate the results and compare with one of the conventional methods used to solve partial differential equations, that is, the compact finite difference method.

Network Graph Laplacian-Based Sensor Projection in System-Level Modeling of Liquid-Metal Loop

Accurate modeling of nuclear reactor transients is important for improving operation and maintenance planning strategies in advanced reactors. An adequate understanding of the connectivity between various components and sensor signals can provide the means to project plant information even where sensors are not operational. This work presents a method for projecting temperature signals using network graph Laplacian and support vector regression (SVR). A system model of a scaled liquid-metal-cooled reactor experiment facility is used to generate thermal-hydraulic data simulating cold-shock transients into the reactor plenum. Two different flow rates are investigated, 0.03 kg/s and 0.06 kg/s, describing the different degrees of nonlinear disturbances in two cases. The temperatures at 72 locations within the system are used to test the temperature prediction model. The inter-connectivity of these points is described using the network graph Laplacian, which describes a location in the loop as a node and then comprises a weighted matrix of connections between each of these nodes. Additional nodes, called ghost nodes, are used to model the heat transfer of the loop with the surroundings. The graph Laplacian and corresponding nodal sensing data are then used to train a kernel, which can inform the impact of various phenomena at one location in the plant on phenomena at other locations in the plant. The sensing data and graph network of these nodes are used to construct a surrogate model of the liquid-metal loop. This model is then used to predict the behavior at certain nodes where the training data are not provided to the model. These model predictions are compared against the test data for the two different inlet flow rates. When optimized, the average error of the simulated temperature data remained below 5% when 3 of the 72 nodes are predicted using the graph-based SVR model. As the number of unknown sensors is increased, the root-mean-square error increases slightly but still remains below 2% when 24 of the 72 nodes are unknown or tested. The test or unknown sensor location plays a larger role than the number of unknowns, with sensor locations near ghost nodes and near the outlet pipes in the plenum having the largest error, with a maximum recorded error of 7%. The objective of the graph-based SVR model is to not only capture the temperature or field variable accurately but also to capture the relative connectivity between the simulated sensors. The correlation coefficient matrix provides a scaled reference for the correlation between temperatures at two different node locations. The calculated correlation coefficient for the simulated temperature data and actual temperature data is within 5% for most of the system, with a maximum relative error of 15%.

Mapping flagellated swimmers to surface-slip driven swimmers

Flagellated microswimmers are ubiquitous in natural habitats. Understanding the hydrodynamic behavior of these cells is of paramount interest, owing to their applications in bio-medical engineering and disease spreading. Since the last two decades, computational efforts have been continuously improved to accurately capture the complex hydrodynamic behavior of these model systems. However, modeling the dynamics of such swimmers with fine details is computationally expensive due to the large number of unknowns and the small time-steps required to solve the equations. In this work we propose a method to map fully resolved flagellated microswimmers to coarse, active slip driven swimmers which can be simulated at a reduced computational cost. Using the new method, the slip driven swimmers move with the same velocity, to machine precision, as the flagellated swimmers and generate a similar flow field with a controlled accuracy. The method is validated for swimming patterns near a no-slip boundary, interactions between swimmers and scattering with large obstacles.

A domain decomposing reduced-order method for solving the multi-domain transient heat conduction problems

In this article, a domain decomposition algorithm combined with a reduced-order model (ROM) is proposed to deal with the transient heat conduction problems in multi-domain structures. Using the domain decomposition algorithm and the finite element method (FEM), the original problem is decomposed into multiple non-overlapping subdomains, which can be solved independently under appropriate boundary conditions. However, additional inter-regional connected calculation will be brought through the simple geometry decomposition. Thus, the proper orthogonal decomposition (POD), a projection-based ROM technique, is utilized to further reduce the number of unknowns and the calculation time for each subdomain. To be more specific, the local POD bases of each subdomain are obtained by decomposing a snapshot matrix that gathered according to time by the application of the virtual boundary conditions on interfaces of the adjacent subdomains and other boundaries. Furthermore, the problems under different boundary conditions can be linked by the variable condensation technique on the interfaces, which enhances the efficiency of the numerical solver dramatically. Finally, the accuracy and efficiency of the proposed method are demonstrated by comparing the results calculated using both the proposed algorithm and the direct finite element simulation.

Numerical Analysis for Sturm–Liouville Problems with Nonlocal Generalized Boundary Conditions

For the generalized Sturm–Liouville problem (GSLP), a new formulation is undertaken to reduce the number of unknowns from two to one in the target equation for the determination of eigenvalue. The eigenparameter-dependent shape functions are derived for using in a variable transformation, such that the GSLP becomes an initial value problem for a new variable. For the uniqueness of eigenfunction an extra condition is imposed, which renders the right-end value of the new variable available; a derived implicit nonlinear equation is solved by an iterative method without using the differential; we can achieve highly precise eigenvalues. For the nonlocal Sturm–Liouville problem (NSLP), we consider two types of integral boundary conditions on the right end. For the first type of NSLP we can prove sufficient conditions for the positiveness of the eigenvalue. Negative eigenvalues and multiple solutions may exist for the second type of NSLP. We propose a boundary shape function method, a two-dimensional fixed-quasi-Newton method and a combination of them to solve the NSLP with fast convergence and high accuracy. From the aspect of numerical analysis the initial value problem of ordinary differential equations and scalar nonlinear equations are more easily treated than the original GSLP and NSLP, which is the main novelty of the paper to provide the mathematically equivalent and simpler mediums to determine the eigenvalues and eigenfunctions.

Coherence-based sufficient condition for support recovery using block generalized orthogonal matching pursuit

Challenge is to find the support vectors of the unknown block sparse vector with compressed measurements in an underdetermined system where the number of unknowns is more than that of measurements. To recover unknown block sparse vector, restricted isometry property (RIP) is a sufficient condition need to be satisfied. Finding the restricted isometric constant is a non-polynomial hard problem for large values of n. In this paper coherence-based recovery guarantee has been proposed to recover the support vectors using block generalized orthogonal matching pursuit (BGOMP). It is proved that BGOMP can able to recover the support vectors with lesser number of iteration than block orthogonal matching pursuit (BOMP) by selecting multiple block support elements per iteration. Simulation results show detection performance of BGOMP is better than BOMP, block subspace pursuit (BSP) and block compressive sampling matching pursuit (BCoSaMP) for different block sparsity and block length. In most of the cases for different block sparsity and block length computation time for BGOMP is lesser than BCoSaMP, BSP and BOMP due to the multiple selection of elements in each iteration.