Simple Computer Implementation Research Articles

EXPRESS CALCULATION OF OXYGEN REGIME IN WASTEWATER FACILITIES

Problem statement. The efficiency of biological water purification depends significantly on the concentration of dissolved oxygen in the volume of the bioreactor. But during the operation of bioreactors, the oxygen regime changes, so it is important to predict the oxygen concentration for different modes of operation of treatment plants. Simplified mathematical models are used to calculate the oxygen regime, but these models are designed for typical bioreactors. An important task is the development of effective multivariate numerical models for evaluating the oxygen regime in reactors of different geometric shapes, which differ from “classical” reactors. The purpose of the article. Development of a CFD model for operational evaluation of the oxygen regime in bioreactors. Methodology. For mathematical modeling of the oxygen regime in the bioreactor, a two-dimensional mass transfer equation is used, which takes into account the place of oxygen supply from the aerator located in the building, the hydrodynamics of the flow in the bioreactor, and the diffusion process. The potential motion model was used to solve the hydrodynamic problem. Numerical integration of modeling equations is carried out by using finite-difference splitting schemes. At each splitting step, the difference equations are solved using explicit schemes. This allows a simple computer implementation of the numerical model. Scientific novelty. A multifactor two-dimensional CFD model was developed, which allows for a quick assessment of the oxygen regime in a bioreactor used for wastewater treatment. Practical significance. The developed CFD model and computer code can be used at the stage of designing or re-engineering bioreactors in order to find the optimal location of aerators in the structure. Conclusions. The software implementation of the developed numerical model was carried out. The results of a computational experiment on the study of the wastewater treatment process in a sedimentation tank are presented.

Hydrodynamics simulation of red blood cells: Employing a penalty method with double jump composition of lower order time integrator

We propose a numerical framework tailored for simulating the dynamics of vesicles with inextensible membranes, which mimic red blood cells, immersed in a non‐Newtonian fluid environment. A penalty method is proposed to handle the inextensibility constraint by relaxation, allowing a simple computer implementation and an affordable computational load compared to the full mixed formulation. To handle the high‐order derivatives in the stress jump across the membrane, which arise due to the high geometric order of the Helfrich functional, we employ higher degree finite elements for spatial discretization. The time integration scheme relies on the double composition of the Crank–Nicolson scheme to achieve faster fourth‐order convergence behavior. Additionally, an adaptive time‐stepping strategy based on a third‐order temporal integration error estimation is implemented. We address the main features of the proposed method, which is benchmarked against existing numerical and experimental results. Furthermore, we investigate the influence of non‐Newtonian rheology on the system dynamics.

Criterion of solvability of the discriminant analysis problem underlying the medical organization management model

The article considers the task of building an effective and efficient organization of the management processes of a medical organization. A distinctive feature is noted - social orientation as the main function, and not just profit maximization with limited resources. This approach requires taking into account not only financial and resource components, but also the impact of many other external factors. However, the authors, realizing the difficulty of describing them, propose to take into account the effect of the "social orientation of the institution" in an implicit form when describing the financial condition of the organization. Taking into account implicit, including external, factors is possible when implementing models not only of statistical and financial analysis, as is usually accepted, but also their extensions to some image recognition methods, which often have a simple computer implementation. In particular, the formalization of the financial condition model of the organization is proposed, which reduces to the problem of mathematical programming with state vectors of both internal and external factors. The possibility of considering a decision-making model that would take into account the influence of possible external factors is also considered, while the "informality" of the problem in the sense of the absence of classical methods of solution is indicated. In this regard, it is proposed to reduce the model to the task of discriminant analysis, which is essentially a system of inequalities. Methods for solving systems of linear inequalities are known, but they are difficult to implement. For this reason, it is proposed to add an optimality criterion (objective function) to the discriminant analysis model, which allows to obtain a linear programming problem that already has simpler computer implementations of solutions. However, this possibility is present only in the case of linear constraints, which is a somewhat special case of the general problem. To solve the problem, in its broadest sense, it is proposed not only to reduce it to mathematical programming, and then to discriminant analysis, but also an effective recognition algorithm indicating the solvability of the problem. The process of constructive problem solving often requires a lot of time and effort, even with the powerful development of modern technologies. With the help of the given recognition algorithm, the discriminant analysis model in some formulation is output in a matrix form, which allows us to conclude when the model has the existence of a separating functional. The authors also propose the use of majority committee methods. The derived algorithm is quite simple to implement, but at the same time effective and efficient in practice, which makes it universal for this kind of tasks.

Quadratic splines on quad-tri meshes: Construction and an application to simulations on watertight reconstructions of trimmed surfaces

Given an unstructured mesh consisting of quadrilaterals and triangles (we allow both planar and non-planar meshes of arbitrary topology), we present the construction of quadratic splines of mixed smoothness — C1 smooth away from the unstructured regions of T and C0 smooth otherwise. The splines have several useful B-spline-like properties – partition of unity, non-negativity, local support and linear independence – and allow for straightforward imposition of boundary conditions. We propose a non-nested refinement process for the splines with multiple advantages — a simple computer implementation, reduction in the footprint of C0 smoothness, boundary preservation, and excellent approximation behaviour in simulations. Furthermore, the refinement process leaves the splines invariant on the mesh boundary. Numerical tests indicate that the spline spaces demonstrate optimal approximation behaviour in the L2 and H1 norms under mesh refinement, and provide a viable approach to simulations on watertight reconstructions of trimmed surfaces.

The nodal-based floating frame of reference formulation with modal reduction

In a recent paper of the authors, a novel nodal-based floating frame of reference formulation (FFRF) for solid finite elements has been proposed. The nodal-based approach bypasses the unhandy inertia shape integrals ab initio, i.e. they neither arise in the derivation nor in the final equations of motion, leading to a surprisingly simple derivation and computer implementation without a lumped mass approximation, which is conventionally employed within commercial multibody codes. However, the nodal-based FFRF has so far been presented without modal reduction, which is usually required for efficient simulations. Hence, the aim of this follow-up paper is to bring the nodal-based FFRF into a suitable form, which allows the incorporation of modal reduction techniques to reduce the overall system size down to the number of modes included in the reduction basis, which further reduces the computational complexity significantly. Moreover, this exhibits a way to calculate the so-called FFRF invariants, which are constant “ingredients” required to set up the FFRF mass matrix and quadratic velocity vector, without integrals and without a lumped mass approximation.

Two-step phase-shifting interferometry for illumination highly non-uniform by fitting 2D polynomials of degree K

This paper presents a robust and non-iterative algorithm for phase retrieval from two interferograms changed by an arbitrary unknown phase-step. First, the object phase is algebraically eliminated; second, the background and modulation light are approximated to 2D polynomials of degree K; third, an error function is algebraically derived; fourth, the least squares method is applied for estimating the phase-step, background, and modulation light, and finally the object phase is retrieved. With this approach, it is not necessary to do additional measurements. The important advantages of this method are its simple computer implementation and its capacity to support high spatial variations in the illumination. This approach is described theoretically, and verified numerically and experimentally.

Self-calibrating generalized phase-shifting interferometry of three steps by fitting modulation light to a polynomial of degree K

This paper presents a robust and non-iterative algorithm for phase retrieval in phase-shifting interferometry of three unknown and unequal phase-steps, based on the modulation light estimation by applying the least squares method (LSM). This approach is divided in three stages; first the background light and phase object are algebraically eliminated to obtain a single equation in three unknowns; second, the square of modulation light is approximated to a polynomial of degree K to build an error function, and third, the coefficients of this polynomial are estimated by applying the LSM, with which a solvable system of equations is formed to calculate the phase-steps, and after that, the phase object is possible to compute. Additional advantages of this approach are its simple computer implementation, and its capacity to support high spatial variations in the illumination and the phase object. This approach is described theoretically and verified numerically and experimentally.

Direct design of 2D RF pulses using matrix inversion

Multi-dimensional pulses are frequently used in MRI for applications such as targeted excitation, fat–water separation or metabolic imaging with hyperpolarised 13C compounds. For the design, the problem is typically separated into the different dimensions. In this work, a method to directly design two-dimensional pulses using the small-tip angle approximation is introduced based on a direct matrix representation. The numerical problem is solved in a single step directly in two dimensions by matrix inversion. Exemplary spectral–spatial excitation and spatio-temporal encoding (SPEN) pulses are designed and validated. The main benefits of the direct design approach include a reduction of artefacts in case of spectral–spatial pulses, a simple and straightforward computer implementation and high flexibility in the pulse design.

Order of Magnitude Scaling: A Systematic Approach to Approximation and Asymptotic Scaling of Equations in Engineering

This work introduces the “order of magnitude scaling” (OMS) technique, which permits for the first time a simple computer implementation of the scaling (or “ordering”) procedure extensively used in engineering. The methodology presented aims at overcoming the limitations of the current scaling approach, in which dominant terms are manually selected and tested for consistency. The manual approach cannot explore all combinations of potential dominant terms in problems represented by many coupled differential equations, thus requiring much judgment and experience and occasionally being unreliable. The research presented here introduces a linear algebra approach that enables unassisted exhaustive searches for scaling laws and checks for their self-consistency. The approach introduced is valid even if the governing equations are nonlinear, and is applicable to continuum mechanics problems in areas such as transport phenomena, dynamics, and solid mechanics. The outcome of OMS is a set of power laws that estimates the characteristic values of the unknowns in a problem (e.g., maximum velocity or maximum temperature variation). The significance of this contribution is that it extends the range of applicability of scaling techniques to large systems of coupled equations and brings objectivity to the selection of small terms, leading to simplifications. The methodology proposed is demonstrated using a linear oscillator and thermocapillary flows in welding.

Numerical valuation of discrete double barrier options

In the present paper we explore the problem for pricing discrete barrier options utilizing the Black–Scholes model for the random movement of the asset price. We postulate the problem as a path integral calculation by choosing approach that is similar to the quadrature method. Thus, the problem is reduced to the estimation of a multi-dimensional integral whose dimension corresponds to the number of the monitoring dates. We propose a fast and accurate numerical algorithm for its valuation. Our results for pricing discretely monitored one and double barrier options are in agreement with those obtained by other numerical and analytical methods in Finance and literature. A desired level of accuracy is very fast achieved for values of the underlying asset close to the strike price or the barriers. The method has a simple computer implementation and it permits observing the entire life of the option.

A simple computer implementation of membrane wrinkle behaviour via a projection technique

AbstractA simple computer implementation of membrane wrinkle behaviour is presented within the classical elastic plane stress constitutive model. In the present method, a projection technique is utilized for modelling of the wrinkle mechanisms, in which the total strains in wrinkled membranes are decomposed into elastic and zero‐strain energy parts, and a projection matrix that extracts the elastic parts from the total strains is derived. The resulting modified elasticity matrix that represents the stress–strain relations in wrinkled membranes is thus obtained as product of the classical elasticity matrix and the projection matrix. The modified elasticity matrix is straightforward to implement within the context of the finite element method. Numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed method. Copyright © 2007 John Wiley & Sons, Ltd.

A second-order finite volume scheme for three dimensional truncated pyramidal quantum dot

Three dimensional truncated pyramidal quantum dots are simulated numerically to compute the energy states and the wave functions. The simulation of the hetero-structures is realized by using a novel finite volume scheme to solve the Schrödinger equation. The simulation benefits greatly from the finite volume scheme in threefold. Firstly, the BenDaniel–Duke hetero-junction interface condition is ingeniously embedded into the scheme. Secondly, the scheme uses uniform meshes in discretization and leads to simple computer implementation. Thirdly, the scheme is efficient as it achieves second-order convergence rates over varied mesh sizes. The scheme has successfully computed all the confined energy states and visualized the corresponding wave functions. The results further predict the relation of the energy states and wave functions versus the height of the truncated pyramidal quantum dots.

Probability maps as a measure of reliability for indivisibility analysis

Digital terrain models (DTMs) represent segments of spatial data bases related to presentation of terrain features and landforms. Square grid elevation models (DEMs) have emerged as the most widely used structure during the past decade because of their simplicity and simple computer implementation. They have become an important segment of Topographic Information Systems (TIS), storing natural and artificial landscape in forms of digital models. This kind of a data structure is especially suitable for morph metric terrain evaluation and analysis, which is very important in environmental and urban planning and Earth surface modeling applications. One of the most often used functionalities of Geographical information systems software packages is indivisibility or view shed analysis of terrain. Indivisibility determination from analog topographic maps may be very exhausting, because of the large number of profiles that have to be extracted and compared. Terrain representation in form of the DEMs databases facilitates this task. This paper describes simple algorithm for terrain view shed analysis by using DEMs database structures, taking into consideration the influence of uncertainties of such data to the results obtained thus far. The concept of probability maps is introduced as a mean for evaluation of results, and is presented as thematic display.

Set-valued maps for image segmentation

The goal of this paper is an analytical basis of a region growing method using set-valued maps. They are to provide an approach to image segmentation that does not use a priori conditions on the final segment. Seizing a suggestion of Demongeot and Leitner ([8]), we start with a compact subset of the grey-valued image, and the region growing method is based on the continuous deformation of sets for decreasing some error functional. Avoiding any further restrictions on these sets leads to describing the process as a set-valued map. The ansatz of the deforming sets utilizes reachable sets of differential inclusions that admit more than one velocity of propagation at each point. So set-valued maps underlie a mathematical segmentation problem posed and solved in the first part. Then we present a suggestion how to apply the analytical results to a very simple computer implementation.

The Remarkable Kaleidoscopic Decompositions of the Bivariate Gaussian Distribution

The construction of bivariate Gaussian distributions by Puente and Klebanoff,1 as transformations of diffuse probability distributions via space-filling fractal interpolating functions is reviewed. It is shown that the ergodic theorem,2 which allows a simple computer implementation of the construction for some diffuse measures, leads in a great many instances, to unexpected kaleidoscopic patterns of exotic nature, which, when added, yield the bivariate Gaussian distribution. Examples of such patterns, some resembling natural shapes, are presented.

Cellular Automata for Elementary Image Enhancement

We study various cellular automata as algorithms for elementary image enhancement, which refers to methods used to improve features of an image without previous information about them that can be implemented by straightforward techniques. Cellular automata appear as natural tools for image processing due to their local nature and simple parallel computer implementation. For this reason various cellular automata algorithms for sharpening and smoothing are presented and studied in this context. Their dynamical behavior is characterized for sequential and parallel updating by associating to them strictly decreasing functionals with the dynamics of the automata. Since tight bounds for the transient time usually cannot be obtained from these operators, a numerical study is needed to analyze their typical performance and effects. For this purpose we compare them with the classical methods for real two dimensional images in terms of convergence rate, effects, and stability in front of noise. The cellular automata methods studied present very fast convergence to fixed points, noise stability, and improvements on real images, which are features that allow us to propose them as a first level elementary image enhancement.

Two-photon bremsstrahlung

We present compact expressions for the matrix elements for double initial-state bremsstrahlung in the process e + e −→ f f at LEP energies. Special attention is given to kinematical situations where one or both of the bremsstrahlung photons are collinear with the incoming electrons: this entails a careful description of the transition region between collinear and noncollinear kinematics. The resulting expressions allow for simple and straightforward computer implementation. We describe a Monte Carlo computation based on these matrix elements, which contains two-loop corrections to the initial state and one-loop corrections to the final state. The results are compared with other analytical and Monte Carlo studies.

Algorithms for Separable Nonlinear Least Squares Problems

Iterative algorithms of Gauss–Newton type for the solution of nonlinear least squares problems are considered. They separate the variables into two sets in such a way that in each iteration, optimization with respect to the first set is performed first, and corrections to those of the second after that. The linear-nonlinear case, where the first set consists of variables that occur linearly, is given special attention, and a new algorithm is derived which is simpler to apply than the variable projection algorithm as described by Golub and Pereyra, and can be performed with no more arithmetical operations than the unseparated Gauss–Newton algorithm. A detailed analysis of the asymptotical convergence properties of both separated and unseparated algorithms is performed. It is found that they have comparable rates of convergence, and all converge almost quadratically for almost compatible problems. Simpler separation schemes, on the other hand, converge only linearly. An efficient and simple computer implementation is described. It uses $QR$ decompositions of appropriate derivative matrices. A series of numerical tests are reported, both on artificial and realistic data. It is found that the theoretical results on asymptotic behavior give a good prediction of the number of iterations needed in a test run. Typical applications are least squares approximation by exponential or rational functions, and overdetermined nonlinear systems of equations; examples are given of both.