Search results for: adjoint

Authors: Mehdi Jafari Matehkolaee

Abstract:

In this paper, we have obtained the adjoint of an arbitrary operator (linear and nonlinear) in Hilbert space by introducing an n-dimensional Riemannian manifold. This general formalism covers every linear operator (non – differential) in Hilbert space. In fact, our approach shows that instead of using the adjoint definition of an operator directly, it can be obtained directly by relying on a suitable generalized space according to the action of the operator in question. For the case of nonlinear operators, we have to change the definition of the linear operator adjoint. But here, we have obtained an adjoint of these operators with respect to the definition of the derivative of the operator. As a matter of fact, we have shown one of the straight applications of the ''Frechet derivative'' in the algebra of the operators.

Keywords: adjoint operator, non-linear operator, differentiable operator, manifold

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Authors: Marcelo Hayashi, João Lima, Bruno Chieregatti, Ernani Volpe

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The adjoint method has been used as a successful tool to obtain sensitivity gradients in aerodynamic design and optimisation for many years. This work presents an alternative approach to the continuous adjoint formulation that enables one to compute gradients of a given measure of merit with respect to control parameters other than those pertaining to geometry. The procedure is then applied to the steady 2–D compressible Euler and incompressible Navier–Stokes flow equations. Finally, the results are compared with sensitivities obtained by finite differences and theoretical values for validation.

Keywords: adjoint method, aerodynamics, sensitivity theory, non-geometric sensitivities

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Authors: Arpit Panwar, Arvind Deshpande

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The study of external vehicle aerodynamics using Spalart-Allmaras turbulence model with adjoint method was conducted. The accessibility and ease of working with the Fluent module of ANSYS and OpenFOAM were considered. The objective of the study was to understand and analyze the possibility of bringing high-level aerodynamic simulation to the average consumer vehicle. A form-factor of BMW M6 vehicle was designed in Solidworks, which was analyzed in OpenFOAM and Fluent. The turbulence model being a single equation provides much faster convergence rate when clubbed with the adjoint method. Fluent being commercial software still does not allow us to solve Spalart-Allmaras turbulence model using the adjoint method. Hence, the turbulence model was solved using the SIMPLE method in Fluent. OpenFOAM being an open source provide flexibility in simulation but is not user-friendly. It supports solving the defined turbulence model with the adjoint method. The result generated from the simulation gives us acceptable values of drag, when validated with the result of percentage error in drag values for a notch-back vehicle model on an extensive simulation produced at 6th ANSA and μETA conference, Greece. The success of this approach will allow us to bring more aerodynamic vehicle body design to all segments of the automobile and not limiting it to just the high-end sports cars.

Keywords: Spalart-Allmaras turbulence model, OpenFOAM, adjoint method, SIMPLE method, vehicle aerodynamic design

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Authors: Elimhan N. Mahmudov

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The paper concerns the necessary and sufficient conditions of optimality for Cauchy problem of fourth order discrete (PD) and discrete-approximate (PDA) inclusions. The main problem is formulation of the fourth order adjoint discrete and discrete-approximate inclusions and transversality conditions, which are peculiar to problems including fourth order derivatives and approximate derivatives. Thus the necessary and sufficient conditions of optimality are obtained incorporating the Euler-Lagrange and Hamiltonian forms of inclusions. Derivation of optimality conditions are based on the apparatus of locally adjoint mapping (LAM). Moreover in the application of these results we consider the fourth order linear discrete and discrete-approximate inclusions.

Keywords: difference, optimization, fourth, approximation, transversality

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Authors: Jurriaan Gillissen

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This work is concerned with stabilizing the numerical integration of the Navier-Stokes equation (NSE), backwards in time. Applications involve the detection of sources of, e.g., sound, heat, and pollutants. Stable reverse numerical integration of parabolic differential equations is also relevant for image de-blurring. While the literature addresses the reverse integration problem of the advection-diffusion equation, the problem of numerical reverse integration of the NSE has, to our knowledge, not yet been addressed. Owing to the presence of viscosity, the NSE is irreversible, i.e., when going backwards in time, the fluid behaves, as if it had a negative viscosity. As an effect, perturbations from the perfect solution, due to round off errors or discretization errors, grow exponentially in time, and reverse integration of the NSE is inherently unstable, regardless of using an implicit time integration scheme. Consequently, some sort of filtering is required, in order to achieve a stable, numerical, reversed integration. The challenge is to find a filter with a minimal adverse affect on the accuracy of the reversed integration. In the present work, we explore an adjoint gradient method (AGM) to achieve this goal, and we apply this technique to two-dimensional (2D), decaying turbulence. The AGM solves for the initial velocity field u0 at t = 0, that, when integrated forward in time, produces a final velocity field u1 at t = 1, that is as close as is feasibly possible to some specified target field v1. The initial field u0 defines a minimum of a cost-functional J, that measures the distance between u1 and v1. In the minimization procedure, the u0 is updated iteratively along the gradient of J w.r.t. u0, where the gradient is obtained by transporting J backwards in time from t = 1 to t = 0, using the adjoint NSE. The AGM thus effectively replaces the backward integration by multiple forward and backward adjoint integrations. Since the viscosity is negative in the adjoint NSE, each step of the AGM is numerically stable. Nevertheless, when applied to turbulence, the AGM develops instabilities, which limit the backward integration to small times. This is due to the exponential divergence of phase space trajectories in turbulent flow, which produces a multitude of local minima in J, when the integration time is large. As an effect, the AGM may select unphysical, noisy initial conditions. In order to improve this situation, we propose two remedies. First, we replace the integration by a sequence of smaller integrations, i.e., we divide the integration time into segments, where in each segment the target field v1 is taken as the initial field u0 from the previous segment. Second, we add an additional term (regularizer) to J, which is proportional to a high-order Laplacian of u0, and which dampens the gradients of u0. We show that suitable values for the segment size and for the regularizer, allow a stable reverse integration of 2D decaying turbulence, with accurate results for more then O(10) turbulent, integral time scales.

Keywords: time reversed integration, parabolic differential equations, adjoint gradient method, two dimensional turbulence

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Authors: Mostafa Shahriari, Theophile Chaumont-Frelet, David Pardo

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Resistivity measurements are used to characterize the Earth’s subsurface. They are categorized into two different groups: (a) those acquired on the Earth’s surface, for instance, controlled source electromagnetic (CSEM) and Magnetotellurics (MT), and (b) those recorded with borehole logging instruments such as Logging-While-Drilling (LWD) devices. LWD instruments are mostly used for geo-steering purposes, i.e., to adjust dip and azimuthal angles of a well trajectory to drill along a particular geological target. Modern LWD tools measure all nine components of the magnetic field corresponding to three orthogonal transmitter and receiver orientations. In order to map the Earth’s subsurface and perform geo-steering, we invert measurements using a gradient-based method that utilizes the derivatives of the recorded measurements with respect to the inversion variables. For resistivity measurements, these inversion variables are usually the constant resistivity value of each layer and the bed boundary positions. It is well-known how to compute derivatives with respect to the constant resistivity value of each layer using semi-analytic or numerical methods. However, similar formulas for computing the derivatives with respect to bed boundary positions are unavailable. The main contribution of this work is to provide an adjoint-based formulation for computing derivatives with respect to the bed boundary positions. The key idea to obtain the aforementioned adjoint state formulations for the derivatives is to separate the tangential and normal components of the field and treat them differently. This formulation allows us to compute the derivatives faster and more accurately than with traditional finite differences approximations. In the presentation, we shall first derive a formula for computing the derivatives with respect to the bed boundary positions for the potential equation. Then, we shall extend our formulation to 3D Maxwell’s equations. Finally, by considering a 1D domain and reducing the dimensionality of the problem, which is a common practice in the inversion of resistivity measurements, we shall derive a formulation to compute the derivatives of the measurements with respect to the bed boundary positions using a 1.5D variational formulation. Then, we shall illustrate the accuracy and convergence properties of our formulations by comparing numerical results with the analytical derivatives for the potential equation. For the 1.5D Maxwell’s system, we shall compare our numerical results based on the proposed adjoint-based formulation vs those obtained with a traditional finite difference approach. Numerical results shall show that our proposed adjoint-based technique produces enhanced accuracy solutions while its cost is negligible, as opposed to the finite difference approach that requires the solution of one additional problem per derivative.

Keywords: inverse problem, bed boundary positions, electromagnetism, potential equation

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Authors: Kunpeng Wang, Hongchun, Wu, Liangzhi Cao, Chuanqi Zhao

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The distributions of homogeneous neutron flux within a node were expanded into a set of analytic basis functions which satisfy the diffusion equation at any point in a triangular-z node for each energy group, and nodes were coupled with each other with both the zero- and first-order partial neutron current moments across all the interfaces of the triangular prism at the same time. Based this method, a code TABFEN has been developed and applied to solve the neutron diffusion equation in a complicated geometry. In addition, after a series of numerical derivation, one can get the neutron adjoint diffusion equations in matrix form which is the same with the neutron diffusion equation; therefore, it can be solved by TABFEN, and the low-high scan strategy is adopted to improve the efficiency. Four benchmark problems are tested by this method to verify its feasibility, the results show good agreement with the references which demonstrates the efficiency and feasibility of this method.

Keywords: analytic basis function expansion method, arbitrary triangular-z node, adjoint neutron flux, complicated geometry

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Authors: Yassir I. Dinar

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We computed the weights of the adjoint action of distinguished sl2-triples in Lie algebra of type Dn using mathematical induction.

Keywords: lie algebra, root systems, representation theory, nilpotent orbits

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Authors: A. Chatel, I. S. Torreguitart, T. Verstraete

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The paper deals with a single point optimization of the LS89 turbine using an adjoint optimization and defining the design variables within a CAD system. The advantage of including the CAD model in the design system is that higher level constraints can be imposed on the shape, allowing the optimized model or component to be manufactured. However, CAD-based approaches restrict the design space compared to node-based approaches where every node is free to move. In order to preserve a rich design space, we develop a methodology to refine the CAD model during the optimization and to create the best parameterization to use at each time. This study presents a methodology to progressively refine the design space, which combines parametric effectiveness with a differential evolutionary algorithm in order to create an optimal parameterization. In this manuscript, we show that by doing the parameterization at the CAD level, we can impose higher level constraints on the shape, such as the axial chord length, the trailing edge radius and G2 geometric continuity between the suction side and pressure side at the leading edge. Additionally, the adjoint sensitivities are filtered out and only smooth shapes are produced during the optimization process. The use of algorithmic differentiation for the CAD kernel and grid generator allows computing the grid sensitivities to machine accuracy and avoid the limited arithmetic precision and the truncation error of finite differences. Then, the parametric effectiveness is computed to rate the ability of a set of CAD design parameters to produce the design shape change dictated by the adjoint sensitivities. During the optimization process, the design space is progressively enlarged using the knot insertion algorithm which allows introducing new control points whilst preserving the initial shape. The position of the inserted knots is generally assumed. However, this assumption can hinder the creation of better parameterizations that would allow producing more localized shape changes where the adjoint sensitivities dictate. To address this, we propose using a differential evolutionary algorithm to maximize the parametric effectiveness by optimizing the location of the inserted knots. This allows the optimizer to gradually explore larger design spaces and to use an optimal CAD-based parameterization during the course of the optimization. The method is tested on the LS89 turbine cascade and large aerodynamic improvements in the entropy generation are achieved whilst keeping the exit flow angle fixed. The trailing edge and axial chord length, which are kept fixed as manufacturing constraints. The optimization results show that the multilevel optimizations were more efficient than the single level optimization, even though they used the same number of design variables at the end of the multilevel optimizations. Furthermore, the multilevel optimization where the parameterization is created using the optimal knot positions results in a more efficient strategy to reach a better optimum than the multilevel optimization where the position of the knots is arbitrarily assumed.

Keywords: adjoint, CAD, knots, multilevel, optimization, parametric effectiveness

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Authors: Didier Auroux, Vladimir Groza

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This work is part of STEEP Marie-Curie ITN project, and it focuses on the identification of unknown parameters of the proposed generic Abrasive WaterJet Milling (AWJM) PDE model, that appears as an ill-posed inverse problem. The necessity of studying this problem comes from the industrial milling applications where the possibility to predict and model the final surface with high accuracy is one of the primary tasks in the absence of any knowledge of the model parameters that should be used. In this framework, we propose the identification of model parameters by minimizing a cost function, measuring the difference between experimental and numerical solutions. The adjoint approach based on corresponding Lagrangian gives the opportunity to find out the unknowns of the AWJM model and their optimal values that could be used to reproduce the required trench profile. Due to the complexity of the nonlinear problem and a large number of model parameters, we use an automatic differentiation software tool (TAPENADE) for the adjoint computations. By adding noise to the artificial data, we show that in fact the parameter identification problem is highly unstable and strictly depends on input measurements. Regularization terms could be effectively used to deal with the presence of data noise and to improve the identification correctness. Based on this approach we present results in 2D and 3D of the identification of the model parameters and of the surface prediction both with self-generated data and measurements obtained from the real production. Considering different types of model and measurement errors allows us to obtain acceptable results for manufacturing and to expect the proper identification of unknowns. This approach also gives us the ability to distribute the research on more complex cases and consider different types of model and measurement errors as well as 3D time-dependent model with variations of the jet feed speed.

Keywords: Abrasive Waterjet Milling, inverse problem, model parameters identification, regularization

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Authors: Radoslaw Pytlak, Damian Suski

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The aim of the paper is to provide a computational tool for planning a haemodialysis process. It is shown that optimization methods can be used to obtain the most effective treatment focused on removing both urea and phosphorus during the process. In order to achieve that, the IV–compartment model of phosphorus kinetics is applied. This kinetics model takes into account a rebound phenomenon that can occur during haemodialysis and results in a hybrid model of the process. Furthermore, vector fields associated with the model equations are such that it is very likely that using the most intuitive objective functions in the planning problem could lead to solutions which include sliding motions. Therefore, building computational tools for solving the problem of planning a haemodialysis process has required constructing numerical algorithms for solving optimal control problems with hybrid systems. The paper concentrates on minimum time control of hybrid systems since this control objective is the most suitable for the haemodialysis process considered in the paper. The presented approach to optimal control problems with hybrid systems is different from the others in several aspects. First of all, it is assumed that a hybrid system can exhibit sliding modes. Secondly, the system’s motion on the switching surface is described by index 2 differential–algebraic equations, and that guarantees accurate tracking of the sliding motion surface. Thirdly, the gradients of the problem’s functionals are evaluated with the help of adjoint equations. The adjoint equations presented in the paper take into account sliding motion and exhibit jump conditions at transition times. The optimality conditions in the form of the weak maximum principle for optimal control problems with hybrid systems exhibiting sliding modes and with piecewise constant controls are stated. The presented sensitivity analysis can be used to construct globally convergent algorithms for solving considered problems. The paper presents numerical results of solving the haemodialysis planning problem.

Keywords: haemodialysis planning process, hybrid systems, optimal control, sliding motion

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Authors: Mohammed Husein Mohammad Rashid

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For a bounded linear operator T acting on a complex infinite dimensional Hilbert space ℋ, we say that T is ∗-class A operator (abbreviation T∈A*) if |T²|≥ |T*|². In this article, we prove the following assertions:(i) we establish some conditions which imply the normality of ∗-class A; (ii) we consider ∗-class A operator T ∈ ℬ(ℋ) with reducing kernel such that TX = XS for some X ∈ ℬ(K, ℋ) and prove the Fuglede-Putnam type theorem when adjoint of S ∈ ℬ(K) is dominant operators; (iii) furthermore, we extend the asymmetric Putnam-Fuglede theorem the class of ∗-class A operators.

Keywords: fuglede-putnam theorem, normal operators, ∗-class a operators, dominant operators

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Authors: Lukas Vierus, Thomas Schuster

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A general framework is examined for dynamic tensor field tomography within an inhomogeneous medium characterized by refraction and absorption, treated as an inverse source problem concerning the associated transport equation. Guided by Fermat’s principle, the Riemannian metric within the specified domain is determined by the medium's refractive index. While considerable literature exists on the inverse problem of reconstructing a tensor field from its longitudinal ray transform within a static Euclidean environment, limited inversion formulas and algorithms are available for general Riemannian metrics and time-varying tensor fields. It is established that tensor field tomography, akin to an inverse source problem for a transport equation, persists in dynamic scenarios. Framing dynamic tensor tomography as an inverse source problem embodies a comprehensive perspective within this domain. Ensuring well-defined forward mappings necessitates establishing existence and uniqueness for the underlying transport equations. However, the bilinear forms of the associated weak formulations fail to meet the coercivity condition. Consequently, recourse to viscosity solutions is taken, demonstrating their unique existence within suitable Sobolev spaces (in the static case) and Sobolev-Bochner spaces (in the dynamic case), under a specific assumption restricting variations in the refractive index. Notably, the adjoint problem can also be reformulated as a transport equation, with analogous results regarding uniqueness. Analytical solutions are expressed as integrals over geodesics, facilitating more efficient evaluation of forward and adjoint operators compared to solving partial differential equations. Certainly, here's the revised sentence in English: Numerical experiments are conducted using a Nesterov-accelerated Landweber method, encompassing various fields, absorption coefficients, and refractive indices, thereby illustrating the enhanced reconstruction achieved through this holistic modeling approach.

Keywords: attenuated refractive dynamic ray transform of tensor fields, geodesics, transport equation, viscosity solutions

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Authors: Haileyesus Tessema Alemneh, Abiyu Enyew Molla, Oluwole Daniel Makinde

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Introduction: Faba bean is one of the most important grown plants worldwide for humans and animals. Several biotic and abiotic elements have limited the output of faba beans, irrespective of their diverse significance. Many faba bean pathogens have been reported so far, of which the most important yield-limiting disease is chocolate spot disease (Botrytis fabae). The dynamics of disease transmission and decision-making processes for intervention programs for disease control are now better understood through the use of mathematical modeling. Currently, a lot of mathematical modeling researchers are interested in plant disease modeling. Objective: In this paper, a deterministic mathematical model for chocolate spot disease (CSD) on faba bean plant with an optimal control model was developed and analyzed to examine the best strategy for controlling CSD. Methodology: Three control interventions, quarantine (u2), chemical control (u3), and prevention (u1), are employed that would establish the optimal control model. The optimality system, characterization of controls, the adjoint variables, and the Hamiltonian are all generated employing Pontryagin’s maximum principle. A cost-effective approach is chosen from a set of possible integrated strategies using the incremental cost-effectiveness ratio (ICER). The forward-backward sweep iterative approach is used to run numerical simulations. Results: The Hamiltonian, the optimality system, the characterization of the controls, and the adjoint variables were established. The numerical results demonstrate that each integrated strategy can reduce the diseases within the specified period. However, due to limited resources, an integrated strategy of prevention and uprooting was found to be the best cost-effective strategy to combat CSD. Conclusion: Therefore, attention should be given to the integrated cost-effective and environmentally eco-friendly strategy by stakeholders and policymakers to control CSD and disseminate the integrated intervention to the farmers in order to fight the spread of CSD in the Faba bean population and produce the expected yield from the field.

Keywords: CSD, optimal control theory, Pontryagin’s maximum principle, numerical simulation, cost-effectiveness analysis

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Authors: A.R. Eskandari, M.R. Eskandari

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A 2D complete identification algorithm for dielectric and multiple objects immersed in air is presented. The employed technique consists of initially retrieving the shape and position of the scattering object using a linear sampling method and then determining the electric permittivity and conductivity of the scatterer using adjoint sensitivity analysis. This inversion algorithm results in high computational speed and efficiency, and it can be generalized for any scatterer structure. Also, this method is robust with respect to noise. The numerical results clearly show that this hybrid approach provides accurate reconstructions of various objects.

Keywords: inverse scattering, microwave imaging, two-dimensional objects, Linear Sampling Method (LSM)

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Authors: Aitor Bilbao, Dragos Axinte, John Billingham

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The inverse problem in Energy Beam (EB) Processes consists of defining the control parameters, in particular the 2D beam path (position and orientation of the beam as a function of time), to arrive at a prescribed solution (freeform surface). This inverse problem is well understood for conventional machining, because the cutting tool geometry is well defined and the material removal is a time independent process. In contrast, EB machining is achieved through the local interaction of a beam of particular characteristics (e.g. energy distribution), which leads to a surface-dependent removal rate. Furthermore, EB machining is a time-dependent process in which not only the beam varies with the dwell time, but any acceleration/deceleration of the machine/beam delivery system, when performing raster paths will influence the actual geometry of the surface to be generated. Two different EB processes, Abrasive Water Machining (AWJM) and Pulsed Laser Ablation (PLA), are studied. Even though they are considered as independent different technologies, both can be described as time-dependent processes. AWJM can be considered as a continuous process and the etched material depends on the feed speed of the jet at each instant during the process. On the other hand, PLA processes are usually defined as discrete systems and the total removed material is calculated by the summation of the different pulses shot during the process. The overlapping of these shots depends on the feed speed and the frequency between two consecutive shots. However, if the feed speed is sufficiently slow compared with the frequency, then consecutive shots are close enough and the behaviour can be similar to a continuous process. Using this approximation a generic continuous model can be described for both processes. The inverse problem is usually solved for this kind of process by simply controlling dwell time in proportion to the required depth of milling at each single pixel on the surface using a linear model of the process. However, this approach does not always lead to the good solution since linear models are only valid when shallow surfaces are etched. The solution of the inverse problem is improved by using a discrete adjoint optimization algorithm. Moreover, the calculation of the Jacobian matrix consumes less computation time than finite difference approaches. The influence of the dynamics of the machine on the actual movement of the jet is also important and should be taken into account. When the parameters of the controller are not known or cannot be changed, a simple approximation is used for the choice of the slope of a step profile. Several experimental tests are performed for both technologies to show the usefulness of this approach.

Keywords: abrasive waterjet machining, energy beam processes, inverse problem, pulsed laser ablation

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Authors: Reza Mohammadi

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Using quartic spline, we develop a method for numerical solution of singularly perturbed two-point boundary-value problems. The purposed method is fourth-order accurate and applicable to problems both in singular and non-singular cases. The convergence analysis of the method is given. The resulting linear system of equations has been solved by using a tri-diagonal solver. We applied the presented method to test problems which have been solved by other existing methods in references, for comparison of presented method with the existing methods. Numerical results are given to illustrate the efficiency of our methods.

Keywords: second-order ordinary differential equation, singularly-perturbed, quartic spline, convergence analysis

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Authors: Dongkyu Lee, Thanh Banh Thien, Soomi Shin

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In the present study, the isogeometric topology optimization is proposed for cracked structures through using Solid Isotropic Material with Penalization (SIMP) as a design model. Design density variables defined in the variable space are used to approximate the element analysis density by the bivariate B-spline basis functions. The mathematical formulation of topology optimization problem solving minimum structural compliance is an alternating active-phase algorithm with the Gauss-Seidel version as an optimization model of optimality criteria. Stiffness and adjoint sensitivity formulations linked to strain energy of cracked structure are proposed in terms of design density variables. Numerical examples demonstrate interactions of topology optimization to structures design with cracks.

Keywords: topology optimization, isogeometric, NURBS, design

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Authors: Aigul Manapova

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We consider an optimal control problem in the higher coefficient of nonlinear equations with a divergent elliptic operator and unbounded nonlinearity, and the Dirichlet boundary condition. The conditions imposed on the coefficients of the state equation are assumed to hold only in a small neighborhood of the exact solution to the original problem. This assumption suggests that the state equation involves nonlinearities of unlimited growth and considerably expands the class of admissible functions as solutions of the state equation. We obtain formulas for the first partial derivatives of the objective functional with respect to the control functions. To calculate the gradients the numerical solutions of the state and adjoint problems are used. We also prove that the gradient of the cost function is Lipchitz continuous.

Keywords: cost functional, differentiability, divergent elliptic operator, optimal control, unbounded nonlinearity

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Authors: Eugene Stepanov, Arcady Ponosov

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To model gene regulatory networks one uses ordinary differential equations with switching nonlinearities, where the initial value problem is known to be well-posed if the trajectories cross the discontinuities transversally. Otherwise, the initial value problem is usually ill-posed, which lead to theoretical and numerical complications. In the presentation, it is proposed to apply the theory of hybrid dynamical systems, rather than switched ones, to regularize the problem. 'Hybridization' of the switched system means that one attaches a dynamic discrete component ('automaton'), which follows the trajectories of the original system and governs its dynamics at the points of ill-posedness of the initial value problem making it well-posed. The construction of the automaton is based on the classification of the attractors of the specially designed adjoint dynamical system. Several examples are provided in the presentation, which support the suggested analysis. The method can also be of interest in other applied fields, where differential equations contain switchings, e.g. in neural field models.

Keywords: hybrid dynamical systems, ill-posed problems, singular perturbation analysis, switching nonlinearities

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Authors: A. Giniatoulline

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We consider the internal oscillations of the ocean which are caused by the gravity force and the Coriolis force, for different models with changeable density, heat transfer, and salinity. Traditionally, the mathematical description of the resonance effect is related to the growing amplitude as a result of input vibrations. We offer a different approach: the study of the relation between the spectrum of the internal oscillations and the properties of the limiting amplitude of the solution for the harmonic input vibrations of the external forces. Using the results of the spectral theory of self-adjoint operators in Hilbert functional spaces, we prove that there exists an explicit relation between the localization of the frequency of the external input vibrations with respect to the essential spectrum of proper inner oscillations and the non-uniqueness of the limiting amplitude. The results may find their application in various problems concerning mathematical modeling of turbulent flows in the ocean.

Keywords: computational fluid dynamics, essential spectrum, limiting amplitude, rotating fluid, spectral theory, stratified fluid, the uniqueness of solutions of PDE equations

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Authors: Y. T. Tsai, Jin H. Huang

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The sound pressure level (SPL) of the moving-coil loudspeaker (MCL) is often simulated and analyzed using the lumped parameter model. However, the SPL of a MCL cannot be simulated precisely in the high frequency region, because the value of cone effective area is changed due to the geometry variation in different mode shapes, it is also related to affect the acoustic radiation mass and resistance. Herein, the paper presents the inverse method which has a high ability to measure the value of cone effective area in various frequency points, also can estimate the MCL electroacoustic parameters simultaneously. The proposed inverse method comprises the direct problem, adjoint problem, and sensitivity problem in collaboration with nonlinear conjugate gradient method. Estimated values from the inverse method are validated experimentally which compared with the measured SPL curve result. Results presented in this paper not only improve the accuracy of lumped parameter model but also provide the valuable information on loudspeaker cone design.

Keywords: inverse problem, cone effective area, loudspeaker, nonlinear conjugate gradient method

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Authors: E. Sener, O. Isik, E. Eroglu, U. Sahin

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The main idea is to solve the system of Maxwell’s equations in accordance with the causality principle to get the energy quantities via Airy functions in a hollow rectangular waveguide. We used the evolutionary approach to electromagnetics that is an analytical time-domain method. The boundary-value problem for the system of Maxwell’s equations is reformulated in transverse and longitudinal coordinates. A self-adjoint operator is obtained and the complete set of Eigen vectors of the operator initiates an orthonormal basis of the solution space. Hence, the sought electromagnetic field can be presented in terms of this basis. Within the presentation, the scalar coefficients are governed by Klein-Gordon equation. Ultimately, in this study, time-domain waveguide problem is solved analytically in accordance with the causality principle. Moreover, the graphical results are visualized for the case when the energy and surplus of the energy for the time-domain waveguide modes are represented via airy functions.

Keywords: airy functions, Klein-Gordon Equation, Maxwell’s equations, Surplus of energy, wave boundary operators

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Authors: Yuanhao Gao, Ping Lin, Kees Weijer

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An optimal control system model is proposed for the cell flow in the process of chick embryo gastrulation in this paper. The target is to determine the cellular interaction forces which are hard to measure. This paper will take an approach to investigate the forces with the idea of the inverse problem. By choosing the forces as the control variable and regarding the cell flow as Stokes fluid, an objective functional will be established to match the numerical result of cell velocity with the experimental data. So that the forces could be determined by minimizing the objective functional. The Lagrange multiplier method is utilized to derive the state and adjoint equations consisting the optimal control system, which specifies the first-order necessary conditions. Finite element method is used to discretize and approximate equations. A conjugate gradient algorithm is given for solving the minimum solution of the system and determine the forces.

Keywords: optimal control model, Stokes equation, conjugate gradient method, finite element method, chick embryo gastrulation

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Authors: Eugene Stepanov, Arkadi Ponossov

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Many mathematical models used in biological and other applications are ill-posed. The reason for that is the nature of differential equations, where the nonlinearities are assumed to be step functions, which is done to simplify the analysis. Prominent examples are switched systems arising from gene regulatory networks and neural field equations. This simplification leads, however, to theoretical and numerical complications. In the presentation, it is proposed to apply the theory of hybrid feedback controls to regularize the problem. Roughly speaking, one attaches a finite state control (‘automaton’), which follows the trajectories of the original system and governs its dynamics at the points of ill-posedness. The construction of the automaton is based on the classification of the attractors of the specially designed adjoint dynamical system. This ‘hybridization’ is shown to regularize the original switched system and gives rise to efficient hybrid numerical schemes. Several examples are provided in the presentation, which supports the suggested analysis. The method can be of interest in other applied fields, where differential equations contain step-like nonlinearities.

Keywords: hybrid feedback control, ill-posed problems, singular perturbation analysis, step-like nonlinearities

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Authors: Eugene Stepanov, Arkadi Ponossov

Abstract:

Ordinary differential equations with switching nonlinearities constitute a very useful tool in many applications. The solutions of such equations can usually be calculated analytically if they cross the discontinuities transversally. Otherwise, one has trajectories that slides along the discontinuity, and the calculations become less straightforward in this case. For instance, one of the problems one faces is non-uniqueness of the sliding modes. In the presentation, it is proposed to apply the theory of hybrid dynamical systems to calculate the solutions that are ‘hidden’ in the discontinuities. Roughly, one equips the underlying switched system with an explicitly designed discrete dynamical system (‘automaton’), which governs the dynamics of the switched system. This construction ‘splits’ the dynamics, which, as it is shown in the presentation, gives uniqueness of the resulting hybrid trajectories and at the same time provides explicit formulae for them. Projecting the hybrid trajectories back onto the original continuous system explains non-uniqueness of its trajectories. The automaton is designed with the help of the attractors of the specially constructed adjoint dynamical system. Several examples are provided in the presentation, which supports the efficiency of the suggested scheme. The method can be of interest in control theory, gene regulatory networks, neural field models and other fields, where switched dynamics is a part of the analysis.

Keywords: hybrid dynamical systems, singular perturbation analysis, sliding modes, switched dynamics

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Authors: Okay Gunes

Abstract:

In this article, for the first time in the literature for this subject we propose a new method for the measuring of well-being inequality through a model composed of superimposing satisfaction waves. The displacement of households’ satisfactory state (i.e. satisfaction) is defined in a satisfaction string. The duration of the satisfactory state for a given period of time is measured in order to determine the relationship between utility and total satisfactory time, itself dependent on the density and tension of each satisfaction string. Thus, individual cardinal total satisfaction values are computed by way of a one-dimensional form for scalar sinusoidal (harmonic) moving wave function, using satisfaction waves with varying amplitudes and frequencies which allow us to measure well-being inequality. One advantage to using satisfaction waves is the ability to show that individual utility and consumption amounts would probably not commute; hence it is impossible to measure or to know simultaneously the values of these observables from the dataset. Thus, we crystallize the problem by using a Heisenberg-type uncertainty resolution for self-adjoint economic operators. We propose to eliminate any estimation bias by correlating the standard deviations of selected economic operators; this is achieved by replacing the aforementioned observed uncertainties with households’ perceived uncertainties (i.e. corrected standard deviations) obtained through the logarithmic psychophysical law proposed by Weber and Fechner.

Keywords: Heisenberg uncertainty principle, superimposing satisfaction waves, Weber–Fechner law, well-being inequality

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Authors: Abu Afree Andalib, M. Mezbah Uddin, M. Rafiur Rahman, M. Abir Hossain, Rajia Sultana Kamol

Abstract:

This paper deals with the analysis of flow past the marine turbine blade which is designed by using the blade element theory and momentum theory for the purpose of using in the field of renewable energy. The designed blade is analyzed for various parameters using FSI module of Ansys. Computational Fluid Dynamics is used for the study of fluid flow past the blade and other fluidic phenomena such as lift, drag, pressure differentials, energy dissipation in water. Finite Element Analysis (FEA) module of Ansys was used to analyze the structural parameter such as stress and stress density, localization point, deflection, force propagation. Fine mesh is considered in every case for more accuracy in the result according to computational machine power. The relevance of design, search and optimization with respect to complex fluid flow and structural modeling is considered and analyzed. The relevancy of design and optimization with respect to complex fluid for minimum drag force using Ansys Adjoint Solver module is analyzed as well. The graphical comparison of the above-mentioned parameter using CFD and FEA and subsequently FSI technique is illustrated and found the significant conformity between both the results.

Keywords: blade element theory, computational fluid dynamics, finite element analysis, fluid-structure interaction, momentum theory

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Authors: Natalia Pacheco Rego

Abstract:

The Liezation functor is a map from the category of Leibniz algebras to the category of Lie algebras, which assigns a Leibniz algebra to the Lie algebra given by the quotient of the Leibniz algebra by the ideal spanned by the square elements of the Leibniz algebra. This functor is left adjoint to the inclusion functor that considers a Lie algebra as a Leibniz algebra. This environment fits in the framework of central extensions and commutators in semi-abelian categories with respect to a Birkhoff subcategory, where classical or absolute notions are relative to the abelianization functor. Classical properties of Leibniz algebras (properties relative to the abelianization functor) were adapted to the relative setting (with respect to the Liezation functor); in general, absolute properties have the corresponding relative ones, but not all absolute properties immediately hold in the relative case, so new requirements are needed. Following this line of research, it was conducted an analysis of central derivations of Leibniz algebras relative to the Liezation functor, called as Lie-derivations, and a characterization of Lie-stem Leibniz algebras by their Lie-central derivations was obtained. In this paper, we present an overview of these results, and we analyze some new properties concerning Lie-central derivations and almost inner Lie-derivations. Namely, a Leibniz algebra is a vector space equipped with a bilinear bracket operation satisfying the Leibniz identity. We define the Lie-bracket by [x, y]lie = [x, y] + [y, x] , for all x, y . The Lie-center of a Leibniz algebra is the two-sided ideal of elements that annihilate all the elements in the Leibniz algebra through the Lie-bracket. A Lie-derivation is a linear map which acts as a derivative with respect to the Lie-bracket. Obviously, usual derivations are Lie-derivations, but the converse is not true in general. A Lie-derivation is called a Lie-central derivation if its image is contained in the Lie-center. A Lie-derivation is called an almost inner Lie-derivation if the image of an element x is contained in the Lie-commutator of x and the Leibniz algebra. The main results we present in this talk refer to the conditions under which Lie-central derivation and almost inner Lie-derivations coincide.

Keywords: almost inner Lie-derivation, Lie-center, Lie-central derivation, Lie-derivation

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Authors: Rony Tawk, Boutros Ghannam, Maroun Nemer

Abstract:

Topology optimization technique presents a potential tool for the design and optimization of structures involved in mass and heat transfer. The method starts with an initial intermediate domain and should be able to progressively distribute the solid and the two fluids exchanging heat. The multi-objective function of the problem takes into account minimization of total pressure loss and maximization of heat transfer between solid and fluid subdomains. Existing methods account for the presence of only one fluid, while the actual work extends optimization distribution of solid and two different fluids. This requires to separate the channels of both fluids and to ensure a minimum solid thickness between them. This is done by adding a third objective function to the multi-objective optimization problem. This article uses density approach where each cell holds two local design parameters ranging from 0 to 1, where the combination of their extremums defines the presence of solid, cold fluid or hot fluid in this cell. Finite volume method is used for direct solver coupled with a discrete adjoint approach for sensitivity analysis and method of moving asymptotes for numerical optimization. Several examples are presented to show the ability of the method to find a trade-off between minimization of power dissipation and maximization of heat transfer while ensuring the separation and continuity of the channel of each fluid without crossing or mixing the fluids. The main conclusion is the possibility to find an optimal bi-fluid domain using topology optimization, defining a fluid to fluid heat exchanger device.

Keywords: topology optimization, density approach, bi-fluid domain, laminar steady state regime, fluid-to-fluid heat exchanger

Procedia PDF Downloads 366