Numerical Methods and Applications

In this note we propose a grid refinement procedure for direction splitting schemes for parabolic problems that can be easily extended to the incompressible Navier-Stokes equations. The procedure is developed to be used in conjunction with a direction splitting time discretization. Therefore, the structure of the resulting linear systems is tridiagonal for all internal unknowns, and only the Schur complement matrix for the unknowns at the interface of refinement has a four diagonal structure. Then the linear system in each direction can be solved either by a kind of domain decomposition iteration or by a direct solver, after an explicit computation of the Schur complement. The numerical results on a manufactured solution demonstrate that this grid refinement procedure does not alter the spatial accuracy of the finite difference approximation and seems to be unconditionally stable.

We consider the initial boundary value problem for the homogeneous heat equation, with homogeneous Dirichlet boundary conditions. By the maximum principle the solution is nonnegative for positive time if the initial data are nonnegative. We study to what extent this property carries over to some piecewise linear finite element discretizations, namely the Standard Galerkin method, the Lumped Mass method, and the Finite Volume Element method. We address both spatially semidiscrete and fully discrete methods.

The signed-particle Monte Carlo method for solving the Wigner equation has made multi-dimensional solutions numerically feasible. The latter is attributable to the concept of annihilation of independent indistinguishable particles, which counteracts the exponential growth in the number of particles due to generation. After the annihilation step, the particles regenerated within each cell of the phase-space should replicate the same information as before the annihilation, albeit with a lesser number of particles. Since the semi-discrete Wigner equation allows only discrete momentum values, this information can be retained with regeneration, however, the position of the regenerated particles in the cell must be chosen wisely. A simple uniform distribution over the spatial domain represented by the cell introduces a ‘numerical diffusion’ which artificially propagates particles simply through the process of regeneration. An optimized regeneration scheme is proposed, which counteracts this effect of ‘numerical diffusion’ in an efficient manner.

In this work, a sensitivity study of the Boltzmann equation describing electron transport in one-dimensional Silicon diodes is performed. We focus on the variability of the model outputs according to the variability of input parameters connected to the geometry, temperature and doping concentration of the device. A number of numerical experiments exploiting the Boltzmann Monte Carlo method have been carried out to compute global sensitivity measures. The most popular variance-based sensitivity analysis approaches, such as the Sobol method and Fourier Amplitude Sensitivity Test (FAST), have been applied. First-order and total sensitivity indices in the context of FAST and Sobol methods have been computed. Furthermore, in order to estimate the interaction effects, the calculation of higher-order sensitivity indices have been performed. Based on the numerical results, we are able to classify the inputs according to their influence over the output variability. This allows a systematic approach to give physical interpretations and insights on the design parameters of a diode, which is hardly accessible otherwise.

The problem of balancing of both systematic and stochastic error is very important when Monte Carlo algorithms are used. A Monte Carlo method for integral equations based on balancing of systematic and stochastic errors is presented. An approach to the problem of controlling the error in non- deterministics methods is presented. The problem of obtaining an optimal ratio between the number of realizations

$$N$$

of the random variable and the mean value

$$k$$

of the number of steps in each random trajectory is discussed. Lower bounds for

$$N$$

and

$$k$$

are provided once a preliminary given error is given. Meaningful numerical examples and experiments are presented and discussed. Experimental and theoretical relative errors are presented. Monte Carlo algorithms with various initial and transition probabilities are compared. An almost optimal Monte Carlo algorithm is discussed and it is proven that it gives more reliable results.

Slot machine RTP optimization problem is usually solved by hand adjustment of the symbols placed on the game reels. By controlling the symbols distribution, it is possible to achieve the desired return to player percent (RTP). Some other parameters can also be adjusted (for example, the free spins frequency or the bonus game frequency). In this paper RTP optimization automation, based on genetic algorithms, is proposed.

A new differential evolution (DE) algorithm with a parallel hierarchical topology (HDE) is proposed. The main goal of the paper is to study how the performance of the algorithm is influenced by the use of parallel migration model. The hierarchical model has several control parameters and the influence of these parameters setting is also studied. The performance of HDE algorithm is compared with non-parallel DE algorithm on CEC2013 benchmark suite. Experimental results show that the HDE outperforms the non-parallel DE algorithm significantly in 27 out of 28 test problems.

Multimemetic algorithms (MMAs) are memetic algorithms that explicitly exploit the evolution of memes, i.e., non-genetic expressions of problem-solving strategies. We consider a class of MMAs in which these memes are rewriting rules whose length can be fixed during the run of the algorithm or self-adapt during the search process. We analyze this self-adaptation in the context of spatially-structured MMAs, namely MMAs in which the population is endowed with a certain topology to which interactions (from the point of view of selection and variation operators) are constrained. For the problems considered, it is shown that panmictic (i.e., non-structured) MMAs are more sensitive to this self-adaptation, and that using variable-length memes seems to be a robust strategy throughout different population structures.

In this paper we propose an Ant Colony Optimization (ACO) algorithm for the partition graph coloring problem (PGCP). Given an undirected graph

$$G =(V,E)$$

, whose nodes are partition into a given number of the node sets. The goal of the PGCP is to find a subset

$$V^* \subset V$$

that contains exactly one node for each cluster and a coloring for

$$V^*$$

so that in the graph induced by

$$V^*$$

, two adjacent nodes have different colors and the total number of used colors is minimal. The performance of our algorithm is evaluated on a common benchmark instances set and the computational results show that compared to a state-of-the-art algorithms, our ACO algorithm achieves solid results in very short run-times.

A

ring tree

is a tree graph with an optional additional edge that closes a unique cycle. Such a cycle is called a

ring

and the nodes on it are called

ring nodes

. The

capacitated ring tree problem

(CRTP) asks for a network of minimal overall edge cost that connects given customers to a depot by ring trees. Ring trees are required to intersect in the depot which has to be either a ring node of degree two in a ring tree or a node of degree one if the ring tree does not contain a ring. Customers are predefined as of

type 1

or

type 2

. The type 2 customers have to be ring nodes, whereas type 1 customers can be either ring nodes or nodes in tree sub-structures. Additionally, optional Steiner nodes are given which can be used as intermediate network nodes if advantageous. Capacity constraints bound both the number of the ring trees as well as the number of customers allowed in each ring tree. In this paper we present first advanced neighborhood structures for the CRTP. Some of them generalize existing concepts for the TSP and the Steiner tree problem, others are CRTP-specific. We also describe models to explore these multi-node and multi-edge exchange neighborhoods in one or more ring trees efficiently. Moreover, we embed these techniques in a heuristic multi-start framework and show that it produces high quality results for small and medium size literature instances.

The present work continues investigations on combination between Adaptive Critic Design (ACD) approach - a gradient-based optimization technique - and a more biologically plausible associative or Hebbian learning. Echo state network (ESN) was used as adaptive critic element that was trained minimizing temporal difference error. While in the previous work the actor was a time profile of the action variable, here investigations are extended to the closed loop (feedback) control scheme. The actor is another ESN network and its inputs are some of the process state variables while its output is the value of the controlled variable. The only trainable connections of the actor - from its reservoir to the readout - are trained to minimize (maximize) the critic output. Comparison between backpropagation of utility approach that is gradient descent algorithm and a Hebbian learning law is made. These two approaches are tested on a task for optimization of a complex nonlinear process for bio-polymer production. The obtained results are compared with respect to the convergence speed as well as to the obtained solution, i.e. reached local optima.

Recent publications suggest that resolving multidimensional tasks where optimisation parameters are hundreds and more faces unusual computational limitation. In the same time optimisation algorithms, which perform well on tasks with low number of dimensions, when are applied to high dimensional tasks require infeasible period of time and computational resources. This article presents a novel investigation on Differential Evolution and Particle Swarm Optimisation with enhanced adaptivity and Free Search applied to 200 dimensional versions of three scalable, global, real-value, numerical tests, which optimal values are dependent on dimensions number and virtually unknown for variety of dimensions. The aim is to: (1) identify computational limitations which numerical methods could face on 200 dimensional tests; (2) identify relations between test complexity and period of time required for tests resolving; (3) discover unknown optimal solutions; (4) identify specific methods’ peculiarities which could support the performance on high dimensional tasks. Experimental results are presented and analysed.

The study proposes a new semi-numerical approach to absorbing boundary problem. The developed method relies on the new formulation of generalized impedance boundary conditions. A distinguishing feature of the approach is the possibility to obtain an analytical solution after performing numerical calculations. The accuracy of the presented model was verified by comparing the simulation results with the exact solution for dipole and loop antennas radiation problem. The examples are based on a finite difference scheme but finite element methods can also be used.

We investigate geometric multigrid methods for solving the large, sparse linear systems which arise in isogeometric discretizations of elliptic partial differential equations. We observe that the performance of standard V-cycle iteration is highly dependent on the spatial dimension as well as the spline degree of the discretization space. Conjugate gradient iteration preconditioned with one V-cycle mitigates this dependence, but does not eliminate it. We perform both classical local Fourier analysis as well as a numerical spectral analysis of the two-grid method to gain better understanding of the underlying problems and observe that classical smoothers do not perform well in the isogeometric setting.

The numerical homogenization of anisotropic linear elastic materials with strongly heterogeneous microstructure is studied. The developed algorithm is applied to the case of two-phase composite material: epoxy resin based nanocomposite incorporating nanoclay Cloisite. The upscaling procedure is described in terms of six auxiliary elastic problems for the reference volume element (RVE). A parallel PCG method is implemented for efficient solution of the arising large-scale systems with sparse, symmetric, and positive semidefinite matrices. Then, the bulk modulus tensor is computed from the upscaled stiffness tensor and its eigenvectors are used to define the transformation matrix. The stiffness tensor of the material is transformed with respect to the principle directions of anisotropy (PDA) which gives a canonical (unique) representation of the material properties. Numerical upscaling results are shown. The voxel microstructure of the two-phase composite material is extracted from a high resolution computed tomography image.

The dynamics of beams that undergo large displacements is analyzed in frequency domain and comparisons between models derived by isogeometric analysis and

$$p$$

-FEM are presented. The equation of motion is derived by the principle of virtual work, assuming Timoshenko’s theory for bending and geometrical type of nonlinearity.

As a result, a nonlinear system of second order ordinary differential equations is obtained. Periodic responses are of interest and the harmonic balance method is applied. The nonlinear algebraic system is solved by an arc-length continuation method in frequency domain.

It is shown that IGA gives better approximations of the nonlinear frequency-response functions than the

$$p$$

-FEM when models with the same number of degrees of freedom are used.

The Wigner formalism provides a convenient formulation of quantum mechanics in the phase space. Deterministic solutions of the Wigner equation are especially needed for problems where phase space quantities vary over several orders of magnitude and thus can not be resolved by the existing stochastic approaches. However, finite difference schemes have been problematic due to the discretization of the diffusion term in this differential equation. A new approach, which uses an integral formulation of the Wigner equation that avoids the problematic differentiation, is shown here. The results of the deterministic method are compared and validated with solutions of the Schrödinger equation. Furthermore, certain numerical aspects pertaining to the demanded parallel implementation are discussed.

Standard explicit schemes for parabolic equations are not very convenient for computing practice due to the fact that they have strong restrictions on the time step. This stability restriction is avoided in some explicit schemes based on explicit-implicit splitting of the problem operator (Saul’yev asymmetric schemes, explicit alternating direction (ADE) schemes, group explicit method). These schemes are unconditionally stable, however, their approximation properties are worse than the usual implicit schemes. These explicit schemes are based on the so-called alternating triangle method and can be considered as factorized schemes in which the problem operator is split into a sum of two triangular operators that are adjoint to each other.

Here we propose a multilevel modification of the alternating triangle method, which demonstrates better properties in terms of accuracy. We also consider explicit schemes of the alternating triangle method for the numerical solution of boundary value problems for systems of equations. The study is based on the general theory of stability (well-posedness) of operator-difference schemes.

In this paper the use of the simple shooting-projection method for solving two-point boundary value problems for second-order ordinary integro-differential equations is proposed. Shooting methods are very suitable for solving such equations numerically, as the integral part of the equation can be evaluated while performing the shooting. The simple shooting-projection method consists of the following steps: First, a guess for the initial condition is made and a forward numerical integration is performed so that an initial value problem solution is obtained, called a shooting trajectory. The shooting trajectory satisfies the left boundary constraint but does not satisfy the right boundary constraint. Next, the shooting trajectory is transformed into a projection trajectory that is an approximate boundary value problem solution. Finally, from the projection trajectory a new initial condition is obtained and the procedure is repeated until convergence, i.e. until the boundary value problem solution is obtained within a prescribed precision.

Some extensive numerical simulations of the atmospheric composition fields in Bulgaria have been recently performed. The US EPA Model-3 system was chosen as a modelling tool. The TNO emission inventory was used as emission input. Special pre-processing procedures are created for introducing temporal profiles and speciation of the emissions. The biogenic emissions of VOC are estimated by the model SMOKE. The numerical experiments have been carried out for different emission scenarios, which makes it possible the contribution of emissions from different source categories to be evaluated. The simulations aimed at constructing of ensemble, comprehensive enough as to provide statistically reliable assessment of the atmospheric composition climate of Bulgaria - typical and extreme features of the special/temporal behavior, annual means and seasonal variations, etc. The present one focuses on the results about fine particulate matter. This is a compound with significant impact on human health, so the interest towards it is recently very big.

A numerical investigation is presented for the seismic analysis of tall reinforced concrete (RC) Civil Engineering structures, which have been degradated due to extreme environmental actions and are strengthened by cable elements. The effects of multiple earthquakes on such RC building frames are computed. Damage indices are estimated in order to compare the seismic response of the structures before and after the retrofit by cable element strengthening, and so to elect the optimum strengthening version.

In this work we present a multi-scale computational framework for evaluation of statistical variability in a molecular based non-volatile memory cell. As a test case we analyse a BULK flash cell with polyoxometalates (POM) inorganic molecules used as storage centres. We focuse our discussions on the methodology and development of our innovative and unique computational framework. The capability of the discussed multi-scale approach is demonstrated by establishing a link between the threshold voltage variability and current-voltage characteristics with various oxidation states of the POMs. The presented simulation framework and methodology can be applied not only to the POM based flash cell but they are also transferrable to the flash cells based on alternative molecules used as a storage media.

The tendency to shift from fossil and nuclear energy sources to renewable energy carriers in Germany leads to the necessity to develop effective energy storage systems. Nowadays, caverns excavated in rock salt formations are recognized as the appropriate places for underground storage of energy in the form of compressed air, hydrogen and natural gas. Accurate design of these underground cavities requires suitable numerical simulations employing proper constitutive models to describe the material behavior of rock salt under various geological conditions. In this paper, a rate dependent model is selected to describe the mechanical response of the rock salt around the cavern. This model is implemented in the finite element code CODE-BRIGHT, then its application in numerical modeling of salt caverns is illustrated. Finally, inverse analysis of the synthetic data is performed to identify the material parameters of the selected model. The applied inverse analysis algorithm employs metamodeling technique in order to reduce the computation time of the parameter identification procedure.

In this research, the effect of sub-system on model response for mechanized tunneling process has been taken into consideration. The main aim of this study is to modify the constitutive parameters in a way that the best agreement between numerical results and measurements is obtained. The sub-system includes supporting pressure at the face of the TBM, contraction along the TBM-shield and grouting pressure in the annular gap. The commercially available finite element code, PLAXIS is adopted to simulate the construction process. The soil behavior during the excavation is numerically reproduced by utilizing Hardening Soil model with small strain stiffness (HSsmall). The constitutive parameters are obtained via sensitivity and back analyses while they have been calibrated based on the real measurement of Western Scheldt tunnel in the Netherlands. Both local and global sensitivity analyses are used to distinguish which parameters are most influencing the soil deformation. Thereafter, the model validation is accomplished by applying different scenarios for face pressure distributions with respect to the slope of the tunnel. In addition, the effect of contraction factor is modified individually or coupled with the variation of grouting pressure. Evaluating the influence of the sub-system is conducted to assess its effects on the model responses and to seek the possibility to decrease the disagreement between the calculated displacement and real measured data.

This paper deals with the computational modeling of the in-plane stretch behavior of knitted textiles. Yarns in textiles are modeled as one-dimensional hyperelastic strings with frictional contact. The model is the limiting case of the three-dimensional contact model, as diameters of the yarns’ cross-sections tend to zero. The model is analyzed theoretically and solved by the finite element method with one-dimensional hyperelastic truss elements, with an extension to frictional point-to-point contact. Numerical results for in-plane loading experiments obtained by this approach are discussed and compared with results of real measurements.

A numerical method for simulation of the deformation and drainage of an axisymmetric film between colliding drops in the presence of inter-phase solute transfer at small capillary and Reynolds numbers and small solute concentration variations is presented. The drops are considered to approach each other under a given interaction force. The hydrodynamic part of the mathematical model is based on the lubrication equations in the gap between the drops and the Stokes equations in the drops, coupled with velocity and stress boundary conditions at the interfaces. Both drop and film solute concentrations, related via mass flux balance across the interfaces, are governed by convection-diffusion equations. These equations for the solute concentration in the drops and the film are solved simultaneously by a semi-implicit finite difference method. Tests and comparisons are performed to show the accuracy and stability of the presented numerical method.

Based on a concept for thresholding of wavelet coefficients, which was addressed in [

8

] and further explored in [

6

,

7

], a method for balancing between non-threshold- and threshold shrinking of wavelet coefficients has emerged. Generalized expo-rational B-splines (GERBS) is a blending type spline construction where local functions at each knot are blended together by

$$C^k$$

-smooth basis functions. Global data fitting can be achieved with GERBS by fitting local functions to the data. One property of the GERBS construction is an intrinsic partitioning of the global data. Compression of the global data set can be achieved by applying the shrinking strategy to the GERBS local functions. In this initial study we investigate how this affects the resulting GERBS geometry.

The high bandwidth of ultra wideband (UWB) radars results in a high spatial resolution, typically a few cm. Thanks to good penetration through materials UWB radars can be very helpful, e.g., also in such situations: through-wall tracking of human beings during security operations, through-rubble localization of motionless persons following an emergency, e.g., earthquake or explosion, through-snow detection of people after an avalanche or through-dress security screening at airports for the detection of non-metal objects etc.

An algorithm for the localization of a point target behind a wall based on the information about times of a signal arrival (TOA) to the receivers will be presented. The 3-dimensional case is considered. We suppose that the permittivity of the considered wall is known. The introduced method may be used for known wall width, and also for the wall width determination.

The determination of the through wall TOA for given point targets, based on Snell’s law, has been considered before by other authors. We solve the inverse problem using the Newton method for the minimization of the least squares objective function.

A concept for shrinking of wavelet coefficients has been presented and explored in a series of articles [

2

–

4

]. The theory and experiments so far suggest a strategy where the shrinking adapts to local smoothness properties of the original signal. From this strategy we employ partitioning of the global signal and local shrinking under smoothness constraints. Furthermore, we benchmark shrinking of local partitions’ wavelet coefficients utilizing a selection of wavelet basis functions. Then we present and benchmark an adaptive partition-based shrinking strategy where the best performing shrinkage is applied to individual partitions, one at a time. Finally, we compare the local and global benchmark results.

In this paper we consider a pricing model that predicts increased implied volatility with minimal assumptions beyond those of the Black-Scholes theory. It is described by systems of nonlinear Black-Scholes equations. We propose a two-grid algorithm that consists of two steps: solving the coupled Partial Differential Equations (PDEs) problem on a coarse grid and then solving a number of decoupled sub-problems on a fine mesh by using the coarse grid solution to linearise each PDE of the system. Numerical experiments illustrate the efficiency of the method and validate the related theoretical analysis.

This paper presents a postprocessing technique applied to second- and fourth-order eigenvalue problems. It has been proved that this approach always ensures asymptotically upper bounds for corresponding eigenvalues. The main goal could be formulated as follows: if nonconforming finite elements are used giving lower bounds of eigenvalues, then the presented algorithm is a simple approach for obtaining two-sided bounds of eigenvalues; if conforming finite elements are used by origin, the postprocessing algorithm gives improved approximations of the eigenvalues, which remain asymptotically greater than the exact ones.

Some different aspects of the method applicability are also discussed. Finally, computer based implementations are presented.

In the recent years, the constriction, analysis and application of nonconforming finite elements have been an active research area. So, for fourth-order elliptic problems conforming finite element methods (FEMs) require

$$C^1-$$

continuity, which usually leads to complicated implementation [

1

]. This drawback could be surmounted by using nonconforming methods. These FEMs have been widely applied in computational engineering and structural mechanics.

This paper deals with rectangular variants of the Morley finite elements [

2

]. Beside Adini nonconforming finite element, they can be used for plates with sides parallel to the coordinate axes, such as rectangular plates.

The applicability of different types of Morley rectangles applied for fourth-order problems is also discussed. Numerical implementation and results applied to plate bending problem illustrate the presented investigation.

In the present paper we propose a family of patterns for hotspot load traffic simulating. The results from computer simulations of the throughput of a crossbar packet switch with these patterns are presented. The necessary computations have been performed on the grid-cluster of IICT-BAS. Our simulations utilize the MiMa-algorithm for non-conflict schedule, specified by the apparatus of Generalized Nets. A numerical procedure for computation of the upper bound of the throughput is utilized. It is shown that the throughput of the MiMa-algorithm with the suggested family of patterns tend to 100 %.

We consider the problem of extremal scattered data interpolation in

$$\mathbb {R}^3$$

. Using our previous work on minimum

$$L_2$$

-norm interpolation curve networks, we construct a bivariate interpolant

$$F$$

with the following properties:

(i)

$$F$$

is

$$G^1$$

-continuous,

(ii)

$$F$$

consists of triangular Bézier surfaces,

(iii)

each Bézier surface satisfies the tetra-harmonic equation

$$\varDelta ^4 F=0$$

. Hence

$$F$$

is an extremum to the corresponding energy functional.

We also discuss the case of convex scattered data in

$$\mathbb {R}^3$$

.