Application of Petrov-Galerkin methods to transient boundary value problems in chemical engineering: adsorption with steep gradients in bidisperse solids

Petrov-Galerkin methods are known to be versatile techniques for the solution of a wide variety of convection-dispersion transport problems, including those involving steep gradients. but have hitherto received little attention by chemical engineers. We illustrate the technique by means of the well-known problem of simultaneous diffusion and adsorption in a spherical sorbent pellet comprised of spherical, non-overlapping microparticles of uniform size and investigate the uptake dynamics. Solutions to adsorption problems exhibit steep gradients when macropore diffusion controls or micropore diffusion controls, and the application of classical numerical methods to such problems can present difficulties. In this paper, a semi-discrete Petrov-Galerkin finite element method for numerically solving adsorption problems with steep gradients in bidisperse solids is presented. The numerical solution was found to match the analytical solution when the adsorption isotherm is linear and the diffusivities are constant. Computed results for the Langmuir isotherm and non-constant diffusivity in microparticle are numerically evaluated for comparison with results of a fitted-mesh collocation method, which was proposed by Liu and Bhatia (Comput. Chem. Engng. 23 (1999) 933-943). The new method is simple, highly efficient, and well-suited to a variety of adsorption and desorption problems involving steep gradients. (C) 2001 Elsevier Science Ltd. All rights reserved.

Solution techniques for transport problems involving steep concentration gradients: application to noncatalytic fluid solid reactions

Some efficient solution techniques for solving models of noncatalytic gas-solid and fluid-solid reactions are presented. These models include those with non-constant diffusivities for which the formulation reduces to that of a convection-diffusion problem. A singular perturbation problem results for such models in the presence of a large Thiele modulus, for which the classical numerical methods can present difficulties. For the convection-diffusion like case, the time-dependent partial differential equations are transformed by a semi-discrete Petrov-Galerkin finite element method into a system of ordinary differential equations of the initial-value type that can be readily solved. In the presence of a constant diffusivity, in slab geometry the convection-like terms are absent, and the combination of a fitted mesh finite difference method with a predictor-corrector method is used to solve the problem. Both the methods are found to converge, and general reaction rate forms can be treated. These methods are simple and highly efficient for arbitrary particle geometry and parameters, including a large Thiele modulus. (C) 2001 Elsevier Science Ltd. All rights reserved.

Analysis of a circular patch antenna radiating in a parallel-plate radial guide

A field matching method is described to analyze a recessed circular cavity radiating into a radial waveguide. Using the wall impedance approach, the analysis is divided into two separate problems of the cavity and its external environment. Based on this analysis, a computer algorithm is developed for determining wall admittances as seen at the edge of the patch in the cavity, the radial admittance matrix for the two-probe feed arrangement, and the input impedance as observed from the coaxial line feeding the cavity. This algorithm is tested against the general-purpose Hewlett-Packard finite-element High Frequency Structure Simulator as well as against measured results. Good agreement in all considered cases is noted.

Computation of the linear elastic properties of random porous materials with a wide variety of microstructure

A finite-element method is used to study the elastic properties of random three-dimensional porous materials with highly interconnected pores. We show that Young's modulus, E, is practically independent of Poisson's ratio of the solid phase, nu(s), over the entire solid fraction range, and Poisson's ratio, nu, becomes independent of nu(s) as the percolation threshold is approached. We represent this behaviour of nu in a flow diagram. This interesting but approximate behaviour is very similar to the exactly known behaviour in two-dimensional porous materials. In addition, the behaviour of nu versus nu(s) appears to imply that information in the dilute porosity limit can affect behaviour in the percolation threshold limit. We summarize the finite-element results in terms of simple structure-property relations, instead of tables of data, to make it easier to apply the computational results. Without using accurate numerical computations, one is limited to various effective medium theories and rigorous approximations like bounds and expansions. The accuracy of these equations is unknown for general porous media. To verify a particular theory it is important to check that it predicts both isotropic elastic moduli, i.e. prediction of Young's modulus alone is necessary but not sufficient. The subtleties of Poisson's ratio behaviour actually provide a very effective method for showing differences between the theories and demonstrating their ranges of validity. We find that for moderate- to high-porosity materials, none of the analytical theories is accurate and, at present, numerical techniques must be relied upon.

A monolithic smoothing-gap algorithm for contact-impact based on the signed distance function

A new algorithm has been developed for smoothing the surfaces in finite element formulations of contact-impact. A key feature of this method is that the smoothing is done implicitly by constructing smooth signed distance functions for the bodies. These functions are then employed for the computation of the gap and other variables needed for implementation of contact-impact. The smoothed signed distance functions are constructed by a moving least-squares approximation with a polynomial basis. Results show that when nodes are placed on a surface, the surface can be reproduced with an error of about one per cent or less with either a quadratic or a linear basis. With a quadratic basis, the method exactly reproduces a circle or a sphere even for coarse meshes. Results are presented for contact problems involving the contact of circular bodies. Copyright (C) 2002 John Wiley Sons, Ltd.

Powder forging of a sintered Al-3.8Cu-1Mg-0.8Si-O.1Sn alloy

The forging characteristics of an Al-Cu-Mg-Si-Sn alloy are examined using it new testing strategy which incorporates a double truncated cone specimen and finite element modelling. This sample geometry produces controlled strain distributions within a single specimen and can readily identify the specific strain required to achieve a specific microstructural event by matching the metallographic data with the strain profiles calculated from finite element software, The friction conditions were determined using the conventional friction ring test, which was evaluated using finite element software. The rheological properties of the alloy, evaluated from compression testing of right cylinders, are similar to the properties of conventional aluminium forgings. A hoop strain develops at the outer diameter of the truncated cones and this leads to pore opening at the outer few millimetres. The porosity is effectively removed when the total strain equals the net compressive strain. The strain profiles that develop in the truncated cones are largely independent of the processing temperature and the strain rate although the strain required for pore closure increases as the forging temperature is reduced. This suggests that the microstructure and the strain rate sensitivity may also be important factors controlling pore behaviour. (C) 2002 Elsevier Science B.V. All rights reserved.

Explicit/implicit partitioning and a new explicit form of the generalized alpha method

Most finite element packages use the Newmark algorithm for time integration of structural dynamics. Various algorithms have been proposed to better optimize the high frequency dissipation of this algorithm. Hulbert and Chung proposed both implicit and explicit forms of the generalized alpha method. The algorithms optimize high frequency dissipation effectively, and despite recent work on algorithms that possess momentum conserving/energy dissipative properties in a non-linear context, the generalized alpha method remains an efficient way to solve many problems, especially with adaptive timestep control. However, the implicit and explicit algorithms use incompatible parameter sets and cannot be used together in a spatial partition, whereas this can be done for the Newmark algorithm, as Hughes and Liu demonstrated, and for the HHT-alpha algorithm developed from it. The present paper shows that the explicit generalized alpha method can be rewritten so that it becomes compatible with the implicit form. All four algorithmic parameters can be matched between the explicit and implicit forms. An element interface between implicit and explicit partitions can then be used, analogous to that devised by Hughes and Liu to extend the Newmark method. The stability of the explicit/implicit algorithm is examined in a linear context and found to exceed that of the explicit partition. The element partition is significantly less dissipative of intermediate frequencies than one using the HHT-alpha method. The explicit algorithm can also be rewritten so that the discrete equation of motion evaluates forces from displacements and velocities found at the predicted mid-point of a cycle. Copyright (C) 2003 John Wiley Sons, Ltd.

Knowledge-based system on optimum design of liquid retaining structures with genetic algorithms

This paper delineates the development of a prototype hybrid knowledge-based system for the optimum design of liquid retaining structures by coupling the blackboard architecture, an expert system shell VISUAL RULE STUDIO and genetic algorithm (GA). Through custom-built interactive graphical user interfaces under a user-friendly environment, the user is directed throughout the design process, which includes preliminary design, load specification, model generation, finite element analysis, code compliance checking, and member sizing optimization. For structural optimization, GA is applied to the minimum cost design of structural systems with discrete reinforced concrete sections. The design of a typical example of the liquid retaining structure is illustrated. The results demonstrate extraordinarily converging speed as near-optimal solutions are acquired after merely exploration of a small portion of the search space. This system can act as a consultant to assist novice designers in the design of liquid retaining structures.

A coupled knowledge-based expert system for design of liquid-retaining structures

This paper describes a coupled knowledge-based system (KBS) for the design of liquid-retaining structures, which can handle both the symbolic knowledge processing based on engineering heuristics in the preliminary synthesis stage and the extensive numerical crunching involved in the detailed analysis stage. The prototype system is developed by employing blackboard architecture and a commercial shell VISUAL RULE STUDIO. Its present scope covers design of three types of liquid-retaining structures, namely, a rectangular shape with one compartment, a rectangular shape with two compartments and a circular shape. Through custom-built interactive graphical user interfaces, the user is directed throughout the design process, which includes preliminary design, load specification, model generation, finite element analysis, code compliance checking and member sizing optimization. It is also integrated with various relational databases that provide the system with sectional properties, moment and shear coefficients and final member details. This system can act as a consultant to assist novice designers in the design of liquid-retaining structures with increase in efficiency and optimization of design output and automated record keeping. The design of a typical example of the liquid-retaining structure is also illustrated. (C) 2003 Elsevier B.V All rights reserved.

Biaxial failure model for fiber reinforced concrete

Steel fiber reinforced concrete (SFRC) is widely applied in the construction industry. Numerical elastoplastic analysis of the macroscopic behavior is complex. This typically involves a piecewise linear failure curve including corner singularities. This paper presents a single smooth biaxial failure curve for SFRC based on a semianalytical approximation. Convexity of the proposed model is guaranteed so that numerical problems are avoided. The model has sufficient flexibility to closely match experimental results. The failure curve is also suitable for modeling plain concrete under biaxial loading. Since this model is capable of simulating the failure states in all stress regimes with a single envelope, the elastoplastic formulation is very concise and simple. The finite element implementation is developed to demonstrate the conciseness and the effectiveness of the model. The computed results display good agreement with published experimental data.

On the induced electric field gradients in the human body for magnetic stimulation by gradient coils in MRI

Prior theoretical studies indicate that the negative spatial derivative of the electric field induced by magnetic stimulation may he one of the main factors contributing to depolarization of the nerve fiber. This paper studies this parameter for peripheral nerve stimulation (PNS) induced by time.-varying gradient fields during MRI scans. The numerical calculations are based on an efficient, quasi-static, finite-difference scheme and an anatomically realistic human, full-body model. Whole-body cylindrical and planar gradient sets in MRI systems and various input signals have been explored. The spatial distributions of the induced electric field and their gradients are calculated and attempts are made to correlate these areas with reported experimental stimulation data. The induced electrical field pattern is similar for both the planar coils and cylindrical coils. This study provides some insight into the spatial characteristics of the induced field gradients for PNS in MRI, which may be used to further evaluate the sites where magnetic stimulation is likely to occur and to optimize gradient coil design.

Comparação de desempenho entre a formulação direta do Método dos Elementos de Contorno com Funções Radiais e o Método dos Elementos Finitos em problemas de Poisson e Helmholtz

O presente trabalho objetiva avaliar o desempenho do MECID (Método dos Elementos de Contorno com Interpolação Direta) para resolver o termo integral referente à inércia na Equação de Helmholtz e, deste modo, permitir a modelagem do Problema de Autovalor assim como calcular as frequências naturais, comparando-o com os resultados obtidos pelo MEF (Método dos Elementos Finitos), gerado pela Formulação Clássica de Galerkin. Em primeira instância, serão abordados alguns problemas governados pela equação de Poisson, possibilitando iniciar a comparação de desempenho entre os métodos numéricos aqui abordados. Os problemas resolvidos se aplicam em diferentes e importantes áreas da engenharia, como na transmissão de calor, no eletromagnetismo e em problemas elásticos particulares. Em termos numéricos, sabe-se das dificuldades existentes na aproximação precisa de distribuições mais complexas de cargas, fontes ou sorvedouros no interior do domínio para qualquer técnica de contorno. No entanto, este trabalho mostra que, apesar de tais dificuldades, o desempenho do Método dos Elementos de Contorno é superior, tanto no cálculo da variável básica, quanto na sua derivada. Para tanto, são resolvidos problemas bidimensionais referentes a membranas elásticas, esforços em barras devido ao peso próprio e problemas de determinação de frequências naturais em problemas acústicos em domínios fechados, dentre outros apresentados, utilizando malhas com diferentes graus de refinamento, além de elementos lineares com funções de bases radiais para o MECID e funções base de interpolação polinomial de grau (um) para o MEF. São geradas curvas de desempenho através do cálculo do erro médio percentual para cada malha, demonstrando a convergência e a precisão de cada método. Os resultados também são comparados com as soluções analíticas, quando disponíveis, para cada exemplo resolvido neste trabalho.

Determinação do momento crítico de flambagem lateral com distorção em vigas mistas contínuas de aço e concreto com perfis de alma senoidal

As vigas mistas de aço e concreto estão sendo largamente utilizadas em construções de edifícios e pontes. Ao se combinar o aço com o concreto obtêm-se estruturas mais econômicas, uma vez que se tira proveito das melhores características de cada material. Nas regiões de momento negativo de uma viga mista contínua, a mesa inferior e parte da alma estão comprimidas, se a alma do perfil não tiver rigidez suficiente para evitar a flexão lateral, ela distorcerá gerando um deslocamento lateral e um giro na mesa comprimida, caracterizando um modo de flambagem denominado flambagem lateral com distorção (FLD). O procedimento de verificação à FLD da EN 1994-1-1:2004 originou o método de cálculo da ABNT NBR 8800:2008, entretanto a EN 1994-1-1:2004 não fornece expressão para o cálculo do momento crítico elástico, enquanto a ABNT NBR 8800:2008 prescreve uma formulação proposta por Roik, Hanswille e Kina (1990) desenvolvida para vigas mistas com perfis de alma plana. Embora as normas prescrevam um método de verificação à FLD para vigas mistas com perfis de alma plana, poucos estudos têm sido feitos sobre esse estado-limite. Além disso, tanto a ABNT NBR 8800:2008 quanto as normas internacionais não abordam perfis de alma senoidal. Neste trabalho, foram implementadas análises de flambagem elástica, com auxílio do software ANSYS 14.0 (2011), em modelos de elementos finitos que retratem o comportamento à FLD de vigas mistas de aço e concreto com perfis de alma plana e senoidal. Os modelos numéricos foram constituídos pelo perfil de aço, por uma mola rotacional que restringe parcialmente o giro da mesa superior e uma restrição ao deslocamento lateral, ao longo de todo o comprimento da viga. Os resultados numéricos são comparados com os obtidos pelas formulações de Roik, Hanswille e Kina (1990) e de Hanswille (2002), adaptadas para levar em consideração a corrugação da alma do perfil de aço. Para avaliação das formulações supracitadas e da consistência da modelagem numérica adotada, o momento crítico elástico foi determinado para vigas mistas com perfis de aço de alma plana. Como resultado, um método para o cálculo do momento crítico elástico de vigas mistas de alma senoidal é proposto.

Multi-scale modeling of the mechanical behavior of polymer-based materials

Polymers have become the reference material for high reliability and performance applications. In this work, a multi-scale approach is proposed to investigate the mechanical properties of polymeric based material under strain. To achieve a better understanding of phenomena occurring at the smaller scales, a coupling of a Finite Element Method (FEM) and Molecular Dynamics (MD) modeling in an iterative procedure was employed, enabling the prediction of the macroscopic constitutive response. As the mechanical response can be related to the local microstructure, which in turn depends on the nano-scale structure, the previous described multi-scale method computes the stress-strain relationship at every analysis point of the macro-structure by detailed modeling of the underlying micro- and meso-scale deformation phenomena. The proposed multi-scale approach can enable prediction of properties at the macroscale while taking into consideration phenomena that occur at the mesoscale, thus offering an increased potential accuracy compared to traditional methods.

Predicting the mechanical behavior of amorphous polymeric materials under strain through multi-scale simulation

Polymeric materials have become the reference material for high reliability and performance applications. However, their performance in service conditions is difficult to predict, due in large part to their inherent complex morphology, which leads to non-linear and anisotropic behavior, highly dependent on the thermomechanical environment under which it is processed. In this work, a multiscale approach is proposed to investigate the mechanical properties of polymeric-based material under strain. To achieve a better understanding of phenomena occurring at the smaller scales, the coupling of a finite element method (FEM) and molecular dynamics (MD) modeling, in an iterative procedure, was employed, enabling the prediction of the macroscopic constitutive response. As the mechanical response can be related to the local microstructure, which in turn depends on the nano-scale structure, this multiscale approach computes the stress-strain relationship at every analysis point of the macro-structure by detailed modeling of the underlying micro- and meso-scale deformation phenomena. The proposed multiscale approach can enable prediction of properties at the macroscale while taking into consideration phenomena that occur at the mesoscale, thus offering an increased potential accuracy compared to traditional methods.