The generalized element method /

"UILU-ENG 78 1738."

Finite elements in structural analysis : a bibliography with abstracts.

Mode of access: Internet.

Remote Control Electrodeposition: Patterning on Substrates without Direct Electrical Connections

Thesis (Ph.D.)--University of Washington, 2016-06

Analytical solutions of linear finite- and small-strain one-dimensional consolidation

Analytical solutions are presented for linear finite-strain one-dimensional consolidation of initially unconsolidated soil layers with surcharge loading for both one- and two-way drainage. These solutions complement earlier solutions for initially unconsolidated soil layers without surcharge and initially normally consolidated soil layers with surcharge. Small-strain solutions for the consolidation of initially unconsolidated soil layers with surcharge loading are also presented, and the relationship between the earlier solutions for initially unconsolidated soil without surcharge and the corresponding small-strain solutions, which was not addressed in the earlier work, is clarified. The new solutions for initially unconsolidated soil with surcharge loading can be applied to the analysis of low stress consolidation tests and to the partial validation of numerical solutions of non-linear finite-strain consolidation. They also clarify a formerly perplexing aspect of finite-strain solution charts first noted in numerical solutions. Copyright (C) 2004 John Wiley Sons, Ltd.

A new relaxation method for roll forming problems

Finite element analysis (FEA) of nonlinear problems in solid mechanics is a time consuming process, but it can deal rigorously with the problems of both geometric, contact and material nonlinearity that occur in roll forming. The simulation time limits the application of nonlinear FEA to these problems in industrial practice, so that most applications of nonlinear FEA are in theoretical studies and engineering consulting or troubleshooting. Instead, quick methods based on a global assumption of the deformed shape have been used by the roll-forming industry. These approaches are of limited accuracy. This paper proposes a new form-finding method - a relaxation method to solve the nonlinear problem of predicting the deformed shape due to plastic deformation in roll forming. This method involves applying a small perturbation to each discrete node in order to update the local displacement field, while minimizing plastic work. This is iteratively applied to update the positions of all nodes. As the method assumes a local displacement field, the strain and stress components at each node are calculated explicitly. Continued perturbation of nodes leads to optimisation of the displacement field. Another important feature of this paper is a new approach to consideration of strain history. For a stable and continuous process such as rolling and roll forming, the strain history of a point is represented spatially by the states at a row of nodes leading in the direction of rolling to the current one. Therefore the increment of the strain components and the work-increment of a point can be found without moving the object forward. Using this method we can find the solution for rolling or roll forming in just one step. This method is expected to be faster than commercial finite element packages by eliminating repeated solution of large sets of simultaneous equations and the need to update boundary conditions that represent the rolls.

Towards a Modeling Environment for Earth Systems Simulations

The developments of models in Earth Sciences, e.g. for earthquake prediction and for the simulation of mantel convection, are fare from being finalized. Therefore there is a need for a modelling environment that allows scientist to implement and test new models in an easy but flexible way. After been verified, the models should be easy to apply within its scope, typically by setting input parameters through a GUI or web services. It should be possible to link certain parameters to external data sources, such as databases and other simulation codes. Moreover, as typically large-scale meshes have to be used to achieve appropriate resolutions, the computational efficiency of the underlying numerical methods is important. Conceptional this leads to a software system with three major layers: the application layer, the mathematical layer, and the numerical algorithm layer. The latter is implemented as a C/C++ library to solve a basic, computational intensive linear problem, such as a linear partial differential equation. The mathematical layer allows the model developer to define his model and to implement high level solution algorithms (e.g. Newton-Raphson scheme, Crank-Nicholson scheme) or choose these algorithms form an algorithm library. The kernels of the model are generic, typically linear, solvers provided through the numerical algorithm layer. Finally, to provide an easy-to-use application environment, a web interface is (semi-automatically) built to edit the XML input file for the modelling code. In the talk, we will discuss the advantages and disadvantages of this concept in more details. We will also present the modelling environment escript which is a prototype implementation toward such a software system in Python (see www.python.org). Key components of escript are the Data class and the PDE class. Objects of the Data class allow generating, holding, accessing, and manipulating data, in such a way that the actual, in the particular context best, representation is transparent to the user. They are also the key to establish connections with external data sources. PDE class objects are describing (linear) partial differential equation objects to be solved by a numerical library. The current implementation of escript has been linked to the finite element code Finley to solve general linear partial differential equations. We will give a few simple examples which will illustrate the usage escript. Moreover, we show the usage of escript together with Finley for the modelling of interacting fault systems and for the simulation of mantel convection.

The programming language python in earth system simulations

-scale vary from a planetary scale and million years for convection problems to 100km and 10 years for fault systems simulations. Various techniques are in use to deal with the time dependency (e.g. Crank-Nicholson), with the non-linearity (e.g. Newton-Raphson) and weakly coupled equations (e.g. non-linear Gauss-Seidel). Besides these high-level solution algorithms discretization methods (e.g. finite element method (FEM), boundary element method (BEM)) are used to deal with spatial derivatives. Typically, large-scale, three dimensional meshes are required to resolve geometrical complexity (e.g. in the case of fault systems) or features in the solution (e.g. in mantel convection simulations). The modelling environment escript allows the rapid implementation of new physics as required for the development of simulation codes in earth sciences. Its main object is to provide a programming language, where the user can define new models and rapidly develop high-level solution algorithms. The current implementation is linked with the finite element package finley as a PDE solver. However, the design is open and other discretization technologies such as finite differences and boundary element methods could be included. escript is implemented as an extension of the interactive programming environment python (see www.python.org). Key concepts introduced are Data objects, which are holding values on nodes or elements of the finite element mesh, and linearPDE objects, which are defining linear partial differential equations to be solved by the underlying discretization technology. In this paper we will show the basic concepts of escript and will show how escript is used to implement a simulation code for interacting fault systems. We will show some results of large-scale, parallel simulations on an SGI Altix system. Acknowledgements: Project work is supported by Australian Commonwealth Government through the Australian Computational Earth Systems Simulator Major National Research Facility, Queensland State Government Smart State Research Facility Fund, The University of Queensland and SGI.

The Mechanics of the drawing of polygonal sections from round at elevated temperature

Economic factors such as the rise in cost of raw materials, labour and power, are compelling manufacturers of cold-drawn polygonal sections, to seek new production routes which will enable the expansion in the varieties of metals used and the inclusion of difficult-to-draw materials. One such method generating considerable industrial interest is the drawing of polygonal sections from round at elevated temperature. The technique of drawing mild steel, medium carbon steel and boron steel wire into octagonal, hexagonal and square sections from round at up to 850 deg C and 50% reduction of area in one pass has been established. The main objective was to provide a basic understanding of the process, with particular emphasis being placed on modelling using both experimental and theoretical considerations. Elevated temperature stress-strain data was obtained using a modified torsion testing machine. Data were used in the upper bound solution derived and solved numerically to predict drawing stress strain, strain-rate, temperature and flow stress distribution in the deforming zone for a range of variables. The success of this warm working process will, of course, depend on the use of a satisfactory elevated temperature lubricant, an efficient cooling system, a suitable tool material having good wear and thermal shock resistance and an efficient die profile design which incorporates the principle of least work. The merits and demerits of die materials such as tungsten carbide, chromium carbide, Syalon and Stellite are discussed, principally from the standpoint of minimising drawing force and die wear. Generally, the experimental and theoretical results were in good agreement, the drawing stress could be predicted within close limits and the process proved to be technically feasible. Finite element analysis has been carried out on the various die geometries and die materials, to gain a greater understanding of the behaviour of these dies under the process of elevated temperature drawing, and to establish the temperature distribution and thermal distortion in the deforming zone, thus establishing the optimum die design and die material for the process. It is now possible to predict, for the materials already tested, (i) the optimum drawing temperature range, (ii) the maximum possible reduction of area per pass, (iii) the optimum drawing die profiles and die materials, (iv) the most efficient lubricant in terms of reducing the drawing force and die wear.

The mechanical behaviour of cylindrical GRP panels

Glass reinforced plastic (GRP) is now an established material for the fabrication of sonar windows. Its good mechanical strength, light weight, resistance to corrosion and acoustic transparency, are all properties which fit it for this application. This thesis describes a study, undertaken at the Royal Naval Engineering College, Plymouth, into the mechanical behaviour of a circular cylindrical sonar panel. This particular type of panel would be used to cover a flank array sonar in a ship or submarine. The case considered is that of a panel with all of its edges mechanically clamped and subject to pressure loading on its convex surface. A comprehensive program of testing, to determine the orthotropic elastic properties of the laminated composite panel material is described, together with a series of pressure tests on 1:5 scale sonar panels. These pressure tests were carried out in a purpose designed test rig, using air pressure to provide simulated hydrostatic and hydrodynamic loading. Details of all instrumentation used in the experimental work are given in the thesis. The experimental results from the panel testing are compared with predictions of panel behaviour obtained from both the Galerkin solution of Flugge's cylindrical shell equations (orthotropic case), and finite element modelling of the panels using PAFEC. A variety of appropriate panel boundary conditions are considered in each case. A parametric study, intended to be of use as a preliminary design tool, and based on the above Galerkin solution, is also presented. This parametric study considers cases of boundary conditions, material properties, and panel geometry, outside of those investigated in the experimental work Final conclusions are drawn and recommendations made regarding possible improvements to the procedures for design, manufacture and fixing of sonar panels in the Royal Navy.

Applications of Reissner's principle to structural dynamics

The analysis and prediction of the dynamic behaviour of s7ructural components plays an important role in modern engineering design. :n this work, the so-called "mixed" finite element models based on Reissnen's variational principle are applied to the solution of free and forced vibration problems, for beam and :late structures. The mixed beam models are obtained by using elements of various shape functions ranging from simple linear to complex cubic and quadratic functions. The elements were in general capable of predicting the natural frequencies and dynamic responses with good accuracy. An isoparametric quadrilateral element with 8-nodes was developed for application to thin plate problems. The element has 32 degrees of freedom (one deflection, two bending and one twisting moment per node) which is suitable for discretization of plates with arbitrary geometry. A linear isoparametric element and two non-conforming displacement elements (4-node and 8-node quadrilateral) were extended to the solution of dynamic problems. An auto-mesh generation program was used to facilitate the preparation of input data required by the 8-node quadrilateral elements of mixed and displacement type. Numerical examples were solved using both the mixed beam and plate elements for predicting a structure's natural frequencies and dynamic response to a variety of forcing functions. The solutions were compared with the available analytical and displacement model solutions. The mixed elements developed have been found to have significant advantages over the conventional displacement elements in the solution of plate type problems. A dramatic saving in computational time is possible without any loss in solution accuracy. With beam type problems, there appears to be no significant advantages in using mixed models.

Computer simulation of form-roll design

Changes in modern structural design have created a demand for products which are light but possess high strength. The objective is a reduction in fuel consumption and weight of materials to satisfy both economic and environmental criteria. Cold roll forming has the potential to fulfil this requirement. The bending process is controlled by the shape of the profile machined on the periphery of the rolls. A CNC lathe can machine complicated profiles to a high standard of precision, but the expertise of a numerical control programmer is required. A computer program was developed during this project, using the expert system concept, to calculate tool paths and consequently to expedite the procurement of the machine control tapes whilst removing the need for a skilled programmer. Codifying the expertise of a human and the encapsulation of knowledge within a computer memory, destroys the dependency on highly trained people whose services can be costly, inconsistent and unreliable. A successful cold roll forming operation, where the product is geometrically correct and free from visual defects, is not easy to attain. The geometry of the sheet after travelling through the rolling mill depends on the residual strains generated by the elastic-plastic deformation. Accurate evaluation of the residual strains can provide the basis for predicting the geometry of the section. A study of geometric and material non-linearity, yield criteria, material hardening and stress-strain relationships was undertaken in this research project. The finite element method was chosen to provide a mathematical model of the bending process and, to ensure an efficient manipulation of the large stiffness matrices, the frontal solution was applied. A series of experimental investigations provided data to compare with corresponding values obtained from the theoretical modelling. A computer simulation, capable of predicting that a design will be satisfactory prior to the manufacture of the rolls, would allow effort to be concentrated into devising an optimum design where costs are minimised.

Kineto-elastodynamic analysis of high-speed four-bar mechanism

This thesis addresses the kineto-elastodynamic analysis of a four-bar mechanism running at high-speed where all links are assumed to be flexible. First, the mechanism, at static configurations, is considered as structure. Two methods are used to model the system, namely the finite element method (FEM) and the dynamic stiffness method. The natural frequencies and mode shapes at different positions from both methods are calculated and compared. The FEM is used to model the mechanism running at high-speed. The governing equations of motion are derived using Hamilton's principle. The equations obtained are a set of stiff ordinary differential equations with periodic coefficients. A model is developed whereby the FEM and the dynamic stiffness method are used conjointly to provide high-precision results with only one element per link. The principal concern of the mechanism designer is the behaviour of the mechanism at steady-state. Few algorithms have been developed to deliver the steady-state solution without resorting to costly time marching simulation. In this study two algorithms are developed to overcome the limitations of the existing algorithms. The superiority of the new algorithms is demonstrated. The notion of critical speeds is clarified and a distinction is drawn between "critical speeds", where stresses are at a local maximum, and "unstable bands" where the mechanism deflections will grow boundlessly. Floquet theory is used to assess the stability of the system. A simple method to locate the critical speeds is derived. It is shown that the critical speeds of the mechanism coincide with the local maxima of the eigenvalues of the transition matrix with respect to the rotational speed of the mechanism.

Electromagnetic interference from variable speed motor drives

A methodology is presented which can be used to produce the level of electromagnetic interference, in the form of conducted and radiated emissions, from variable speed drives, the drive that was modelled being a Eurotherm 583 drive. The conducted emissions are predicted using an accurate circuit model of the drive and its associated equipment. The circuit model was constructed from a number of different areas, these being: the power electronics of the drive, the line impedance stabilising network used during the experimental work to measure the conducted emissions, a model of an induction motor assuming near zero load, an accurate model of the shielded cable which connected the drive to the motor, and finally the parasitic capacitances that were present in the drive modelled. The conducted emissions were predicted with an error of +/-6dB over the frequency range 150kHz to 16MHz, which compares well with the limits set in the standards which specify a frequency range of 150kHz to 30MHz. The conducted emissions model was also used to predict the current and voltage sources which were used to predict the radiated emissions from the drive. Two methods for the prediction of the radiated emissions from the drive were investigated, the first being two-dimensional finite element analysis and the second three-dimensional transmission line matrix modelling. The finite element model took account of the features of the drive that were considered to produce the majority of the radiation, these features being the switching of the IGBT's in the inverter, the shielded cable which connected the drive to the motor as well as some of the cables that were present in the drive.The model also took account of the structure of the test rig used to measure the radiated emissions. It was found that the majority of the radiation produced came from the shielded cable and the common mode currents that were flowing in the shield, and that it was feasible to model the radiation from the drive by only modelling the shielded cable. The radiated emissions were correctly predicted in the frequency range 30MHz to 200MHz with an error of +10dB/-6dB. The transmission line matrix method modelled the shielded cable which connected the drive to the motor and also took account of the architecture of the test rig. Only limited simulations were performed using the transmission line matrix model as it was found to be a very slow method and not an ideal solution to the problem. However the limited results obtained were comparable, to within 5%, to the results obtained using the finite element model.

A procedure for the reconstruction of a stochastic stationary temperature field

An iterative procedure is proposed for the reconstruction of a temperature field from a linear stationary heat equation with stochastic coefficients, and stochastic Cauchy data given on a part of the boundary of a bounded domain. In each step, a series of mixed well-posed boundary-value problems are solved for the stochastic heat operator and its adjoint. Well-posedness of these problems is shown to hold and convergence in the mean of the procedure is proved. A discretized version of this procedure, based on a Monte Carlo Galerkin finite-element method, suitable for numerical implementation is discussed. It is demonstrated that the solution to the discretized problem converges to the continuous as the mesh size tends to zero.

Approximation Classes for Adaptive Methods

* This work has been supported by the Office of Naval Research Contract Nr. N0014-91-J1343, the Army Research Office Contract Nr. DAAD 19-02-1-0028, the National Science Foundation grants DMS-0221642 and DMS-0200665, the Deutsche Forschungsgemeinschaft grant SFB 401, the IHP Network “Breaking Complexity” funded by the European Commission and the Alexan- der von Humboldt Foundation.