Set Of Simultaneous Algebraic Equations Research Articles

Soil–pile interaction in vertical vibration in inhomogeneous soils

AbstractThis paper develops a novel reference analytical solution for axially loaded piles in inhomogeneous soils, extending the pioneering elastodynamic model of Nogami and Novak (1976) to piles embedded in vertically inhomogeneous soils. Following the classical earlier model, the pile is modelled as a rod, using the strength‐of‐materials solution, and the soil layer as an approximate continuum, which rest on rigid rock. The approximation lies in reducing the number of dependent variables by eliminating certain stresses and displacements in the governing elastodynamic equations: the vertical normal and vertical shear stresses in the soil are controlled exclusively by the vertical component of the soil displacement. Soil inhomogeneity is introduced via a power law variation of shear modulus with depth, and perfect bonding is assumed at the soil–pile interface. The proposed generalized formulation treats two types of inhomogeneity by employing pertinent eigen expansions of the dependent variables over the vertical coordinate. The response is expressed in terms of generalized Fourier series and includes: (i) displacements and stresses along the pile and the pile–soil interface; and (ii) displacement and stress in the soil. Contrary to available models for homogeneous soils, the associated Fourier coefficients are coupled, obtained as solutions to a set of simultaneous algebraic equations of equal rank to the number of modes considered.

An analytical continuum model for axially loaded end‐bearing piles in inhomogeneous soil

SummaryAn approximate static solution is derived for the elastic settlement and load‐transfer mechanism in axially loaded end‐bearing piles in inhomogeneous soil obeying a power law variation in shear modulus with depth. The proposed generalised formulation can handle different types of soil inhomogeneity by employing pertinent eigenexpansions of the dependent variables over the vertical coordinate, in the form of static soil “modes”, analogous to those used in structural dynamics. Contrary to available models for homogeneous soil, the associated Fourier coefficients are coupled, obtained as solutions to a set of simultaneous algebraic equations of equal rank to the number of modes considered. Closed‐form solutions are derived for the (1) pile head stiffness; (2) pile settlement, axial stress, and side friction profiles leading to actual, depth‐dependent Winkler moduli, (3) displacement and stress fields in the soil; and (4) average, depth‐independent Winkler moduli to match pile head settlement. The predictive power of the model is verified via comparisons against finite element analyses. The applicability to inhomogeneous soil of an existing regression formula for the average Winkler modulus is explored.

Mixed discrete least squares meshless method for solving the linear and non-linear propagation problems

A Mixed formulation of Discrete Least Squares Meshless (MDLSM) as a truly meshfree method is presented in this paper for solving both linear and non-linear propagation problems. In DLSM method, the irreducible formulation was deployed which needs to calculate the costly second derivatives of the MLS shape functions. In the proposed MDLSM method, the complex and costly second derivatives of shape functions are not required. Furthermore, using the mixed formulation, both unknown parameters and their gradients are simultaneously obtained circumventing the need for post-processing procedure performed in irreducible formulation to calculate the gradients. Therefore, the accuracy of gradients of unknown parameters is increased. In MDLSM method, the set of simultaneous algebraic equations are built by minimizing a least squares functional with respect to the nodal parameters. The least squares functional defined as the sum of squared residuals of the differential equation and its boundary condition. The proposed method automatically leads to symmetric and positive-definite system of equations and, therefore, is not subject to the Ladyzenskaja-Babuska-Brezzi (LBB) condition. The proposed MDLSM method is validated and verified by a set of benchmark problems. The results indicate the ability of proposed method to efficiently and effectively solve the linear and non-linear propagation problems.

Feasibility of a compact heat recovery ventilator module with an integrated air-cooled solar absorption air-conditioner

A solar absorption air-conditioner is simulated with an air-to-air heat exchanger to evaluate the feasibility of a compact solar air-conditioning ventilator module. The air-to-air heat exchanger considered in this study is a membrane type total exchanger and the absorption air-conditioner is a single-effect LiBr-water machine with air-coil heat exchangers. All components are modeled in effectiveness-NTU methods including a dehumidifying DX evaporator coil and a cross-flow absorber allowing the whole system to be described by a set of simultaneous algebraic equations, which are then solved easily by a matrix solver. It is predicted that the baseline air-conditioner would produce cooling power in 1.4–5 kW from hot water in 50–100 °C with a COP thrm over 0.7 without the risk of crystallization and that the heat transfer coefficients of the air-coils and the pressure losses would greatly influence the performance. Total cooling power of the baseline system at 80 °C hot water temperature condition is found 19.2 kW, of which 15.7 kW is attributable to AHX and 3.5 kW to the absorption air-conditioner. Corresponding total COP elec is 76, to which the contributions of AHX and the absorption air-conditioner are 62 and 14, respectively. Air flow rates are found to greatly influence the overall performance and should be carefully chosen. It is concluded that the proposed idea is technically feasible and worth further development as an alternative solution.

Analytical modeling of sound transmission through clamped triple-panel partition separated by enclosed air cavities

Abstract An analytical model for sound transmission through a clamped triple-panel partition of finite extent and separated by two impervious air cavities is formulated. The solution derived from the model takes the form of that for a clamp supported rectangular plate. A set of modal functions (or more strictly speaking, the basic functions) are employed to account for the clamped boundary conditions, and the application of the virtual work principle leads to a set of simultaneous algebraic equations for determining the unknown modal coefficients. The sound transmission loss (STL) of the triple-panel partition as a function of excitation frequency is calculated and compared with that of a double-panel partition. The model predictions are then used to explore the physical mechanisms associated with the various dips on the STL versus frequency curve, including the equivalent ‘mass-spring’ resonance, the standing-wave resonance and the panel modal resonance. The asymptotic variation of the solution from a finite-sized partition to an infinitely large partition is illustrated in such a way as to demonstrate the influence of the boundary conditions on the soundproofing capability of the partition. In general, a triple-panel partition outperforms a double-panel partition in insulating the incident sound, and the relatively large number of system parameters pertinent to the triple-panel partition in comparison with that of the double-panel partition offers more design space for the former to tailor its noise reduction performance.

A hydrodynamic network model for interdigitated flow fields

The flow field is one of the main components of a fuel cell, which distributes the reactants to the active area of the cell and evacuates the products formed. Interdigitated flow field (IFF) is one among the different types of flow field designs that forces the reactants or products to flow through the electrode, thereby increasing the cell performance by decreasing concentration polarization loss, however, at the cost of higher-pressure drop. Prior understanding of the reactant and water vapour distribution in a flow field helps in obtaining the best flow field design. In the present paper, a model for the flow distribution and the pressure drop in an IFF has been developed using the analogy between fluid flow and electrical network in which the pressure is made analogous to the voltage and the flow rate to the current. The model, which ultimately reduces to the solution of a set of simultaneous algebraic equations, is capable of predicting the flow split among a set of inlet and outlet channels of an interdigitated flow field as well as the overall pressure drop for laminar, turbulent and two-phase flow conditions for arbitrary number of parallel channels. The results from the hydrodynamic network model have been validated against CFD simulations. This model can therefore be used for the optimization of interdigitated flow field design.

Discrete least squares meshless method with sampling points for the solution of elliptic partial differential equations

A meshfree method namely, discrete least squares meshless (DLSM) method, is presented in this paper for the solution of elliptic partial differential equations. In this method, computational domain is discretized by some nodes and the set of simultaneous algebraic equations are built by minimizing a least squares functional with respect to the nodal parameters. The least squares functional is defined as the sum of squared residuals of the differential equation and its boundary condition calculated at a set of points called sampling points, generally different from nodal points. A moving least squares (MLS) technique is used to construct the shape functions. The proposed method automatically leads to symmetric and positive-definite system of equations. The proposed method does not need any background mesh and, therefore, it is a truly meshless method. The solutions of several one- and two-dimensional examples of elliptic partial differential equations are presented to illustrate the performance of the proposed method. Sensitivity analysis on the parameters of the method is also carried out and the results are presented.

A new BEM technique for nonlinear 2D magnetostatics

The paper presents a new boundary element technique for the solution of nonlinear 2D magnetostatic problems. It is based on a recurrent formulation of Poisson's equation where the change in material property between the domains is modelled as a single layer along the interfaces. Then the nonhomogeneity appears as a source term. The nonlinear regions are decomposed into subdomains where the material is assumed homogeneous. The underlying principle lies on the transmission-line modelling (TLM) method that is commonly used for electric network analysis. A new numerical scheme is derived where the solution is obtained iteratively without any system of algebraic equations for open boundary problems. This is an important point with respect to the classical FEM and BEM treatment that always require the assembly and the solution of a set of simultaneous algebraic equations. Numerical results show significant reduction in memory requirements for our method due to the absence of any matrix storage.

A new solution technique for open nonlinear magnetostatic problems

The paper presents a new FEM-BEM technique for the solution of open nonlinear two-dimensional magnetostatic problems. It is based on a boundary integral formulation that leads to a numerical scheme where the field solution is obtained iteratively without any system of algebraic equations. This is an important point with respect to the treatments that always require the assembly and the solution of a set of simultaneous algebraic equations. Numerical results show significant reduction in memory requirements for our method due to the absence of any matrix storage.

Density matrix theory for calculating magnetization transfer and dynamic lineshape effects

AbstractNMR is a powerful tool for the study of chemical rate processes and the mechanisms by which they occur. Dynamic processes give rise to characteristic changes in transverse and longitudinal relaxation, which manifest themselves in the NMR lineshape and in magnetization transfer phenomena, respectively. Analysis of these effects permits the extraction of kinetic data. The mathematical theory central to this analysis is illustrated using the formalism of the density matrix for a two‐spin system undergoing mutual intramolecular exchange in isotropic solution. The spin Hamiltonian and exchange operators are introduced, and the quantum mechanical equation of motion of the density matrix is presented, emphasizing those density matrix elements that correspond classically to longitudinal and transverse magnetization. The necessary Liouville and exchange superoperators are explicitly constructed from the corresponding Hamiltonian and exchange operators using tensor algebra. The equation of motion is then solved for the populations and single‐quantum coherences to illustrate the origin of magnetization transfer and dynamic lineshape effects. The extension of the theory to arbitrarily large exchanging spin systems is addressed and illustrated for two mutually exchanging spins scalar coupled to a third spin. Finally, the two‐spin lineshape calculation is fully worked out using a steady‐state method that reduces the problem to the solution of a set of simultaneous algebraic equations.

General analytic solution of dynamic response of beams with nonhomogeneity and variable cross-section

In this paper, a new method, the step-reduction method, is proposed to investigate the dynamic response of the Bernoulli-Euler beams with arbitrary nonhomogeneity and arbitrary variable cross-section under arbitrary loads. Both free vibration and forced vibration of such beams are studied. The new method requires to discretize the space domain into a number of elements. Each element can be treated as a homogeneous one with uniform thickness. Therefore, the general analytical solution of homogeneous beams with uniform cross-section can be used in each element. Then, the general analytic solution of the whole beam in terms of initial parameters can be obtained by satisfying the physical and geometric continuity conditions at the adjacent elements. In the case of free vibration, the frequency equation in analytic form can be obtained, and in the case of forced vibration, a final solution in analytical form can also be obtained which is involved in solving a set of simultaneous algebraic equations with only two unknowns which are independent of the numbers of elements divided. The present analysis can also be extended to the study of the vibration of such beams with viscous and hysteretic damping and other kinds of beams and other structural elements with arbitrary nonhomogeneity and arbitrary variable thickness.

Construction of higher order symplectic integrators

Abstract For Hamiltonian systems of the form H = T(p)+V(q) a method is shown to construct explicit and time reversible symplectic integrators of higher order. For any even order there exists at least one symplectic integrator with exact coefficients. The simplest one is the 4th order integrator which agrees with one found by Forest and by Neri. For 6th and 8th orders, symplectic integrators with fewer steps are obtained, for which the coefficients are given by solving a set of simultaneous algebraic equations numerically.

Algebraic method for the design of output deadbeat controller

Based on the deadbeat model approach and the technique of output spectra matching, an algebraic method for the design of an approximate deadbeat controller with a specified configuration has been proposed. The chosen controller is determined by solving a set of simultaneous algebraic equations. One example is given to show the characteristic features of this approach.

Algebraic method for the design of output deadbeat controller

Based on the deadbeat model approach and the technique of output spectra matching, an algebraic method for the design of an approximate deadbeat controller with a specified configuration has been proposed. The chosen controller is determined by solving a set of simultaneous algebraic equations. One example is given to show the characteristic features of this approach.

Trigonometric identities using frequency domain techniques: With practice, it is easier to derive most trigonometric identities than it is to look them up

The engineer is always learning procedures and techniques that make the solving of certain types of problems easier. These techniques often use simplification procedures that not only simplify the analysis, but also transform the problem so that it can be solved using familiar techniques. A good example of this is the solution of electrical circuits where the sources are sinusoidal. The set of integral-differential equations describing the circuit is transformed into a set of simultaneous algebraic equations involving complex numbers. The writing and solving of these equations deal with techniques that are already a familiar part of the engineer's bag-of-tricks. The circuit is transformed into what is called the “frequency domain,” and the emphasis is transformed into one of learning how to deal with complex numbers and complex functions.

The simultaneous use of tracers for ocean circulation studies

A method for the simultaneous use of tracers for oceanic circulation studies is developed. To permit the inclusion of tracers that are subject to biochemical transformation a simple model for photosynthesis, bacterial decomposition and chemical dissolution is formulated based on the use of Redfield ratios for the chemical compounds interaction. The finite difference analogue to the steady state tracer continuity equation, applied to the tracers chosen, leads to a set of simultaneous algebraic equations that in principle can be solved by matrix inversion methods. Two cases are considered: (1) The problem is indeterminate, which may be the case when too few tracers are available and/or some tracer equations are redundant. A solution is obtained by minimizing the transport vector; (2) The problem is incompatible, which may result from making use of many tracers. A solution is obtained by minimizing the errors in satisfying the tracer continuity equations. The method is applied to a 12-box model of the world oceans by considering the surface water, intermediate water, deep water and bottom water of the Arctic, Atlantic, Antarctic and Pacifichndian Oceans (with some simplification for the polar seas). Total dissolved inorganic carbon, alkalinity, phosphorus, oxygen and radiocarbon are used as tracers. A series of steady state solutions are presented and the uncertainties as dependant on box configuration and erroneous or nonrepresentative data are analysed. In the reference solution, circulation cells emerge where Antarctic surface water penetrates into intermediate levels of the two major oceans, from where it undergoes lifting and, at the surface, advection back towards the south. In the Atlantic, much of the penetrating water reaches as far north as to involve the Arctic deep water, which sinks and moves towards the deep Antarctic Ocean through the deep Atlantic. The total upwelling in the Antarctic Ocean is about 30 Sv. Turbulent exchange prevails between surface and intermediate waters in both major oceans with an average K, of about 1 cm 2 s -1 . Particular consideration is given to the transfer of carbon by computing the changes in total carbon and radiocarbon that are induced by fossil fuel combustion and nuclear bomb testing by using the steady state circulation and turbulent transfer deduced. It is concluded that better spatial resolution is required to determine the role of the ocean as a sink for injections of carbon and radiocarbon. DOI: 10.1111/j.1600-0889.1983.tb00025.x

Quantitative predictions of dump combustor flowfields

An approximate solution is proposed for a coaxial dump combustor flowfield. The core region of the flow is treated with the recirculation region shape specified. The problem is thereby reduced to a set of parabolic differential equations. The core region is then divided into a number of streamtubes. This reduces the governing equations to a set of simultaneous algebraic equations which are much simpler to solve. Comparisons with experimental data are presented.

Slurry Flow in Pipe Networks

The basic equations for steady-state flow of a solid-liquid slurry in a network of pipes are developed. The equations are expressed in terms of unknown flow rates and solids concentrations for each pipe in the network. Two sets of simultaneous algebraic equations are obtained. These are nonlinear and a direct solution of the equation sets is not possible. A procedure for solving the equations based on linearization of nonlinear terms is descirbed. This procedure requires several trials and converges rapidly. A computer program was developed based on the proposed method and an example system is analyzed using this program and the results are presented.

Inception of Cavitation From a Solid Surface

This study of cavitation inception from a solid surface includes the presentation of an empirical model for the nuclei, a completely new design for experimental test equipment, a rigorous analysis of liquid conditions within the test section, and quantitative test data. The results prove that it is possible to reduce the effective size of cavitation nuclei, thereby making the interface more resistant to cavitation. The empirical model for cavitation nuclei provides the mechanism for bubble formation from gas-filled, submicroscopic crevices in a solid surface. The size of these nuclei may be reduced by forcing the liquid into these crevices. If the liquid-solid interface is hydrophilic, the pressure treatment is permanent, but the liquid will pull out of a hydrophobic interface thus providing only a temporary reduction of nuclei dimensions. The newly designed experimental equipment for cavitation inception studies has been demonstrated to produce cavitation under controlled conditions. The test section resembles a 180-deg journal bearing constructed in three slices. The middle slice is slid away from the journal to produce a diverging wedge which induces a tensile stress in the liquid. The inception stage of cavitation is detected by an acoustical probe. The temperature and pressure distributions in the thin film are calculated numerically by successively solving the energy and generalized Reynolds equations. A strongly implicit iteration algorithm is employed to solve the large set of simultaneous algebraic equations resulting from placing the nonlinear elliptic partial differential equations in finite difference form. Experimental data from test runs with petroleum oil, glycerin, and a glycerin-soap mixture showed that stress required for inception of cavitation could be determined with a repeatability of ±1.4 × 104 Pa (±2 psi). A linear relationship was experimentally determined to exist between the magnitude of pressure forcing oil into the crevices of a steel surface and a capacity of the oil-steel interface to withstand a tensile stress.

Applications oriented algorithms

Numerical analysis has become an integral part of practical engineering problem analysis. Thus current structural analysis techniques essentially consist of two distinct numerical procedures. Firstly the continuum is discretized by the finite element method, a variant of the well-known Ritz's process, yielding sets of simultaneous algebraic equations for equilibrium and eigenvalue problems or sets of simultaneous ordinary differential equations for the propagation problems. Second such equations are next solved by relevant numerical techniques.