Structural analysis of axisymmetric solids

N the aerospace industry, the stress analysis of complex axisymmetric structures of arbitrary shape subjected to thermal and mechanical loads is of considerable interest. Rocket nozzles and cases, solid-propellant grains, and spacecraft heat shields are practical examples of such structures. Although the governing differential equations for solids of revolution have been known for many years, closed form solutions have been obtained for only a limited number of structures; thus, the stress analyst must rely on experimental or numerical techniques. The finite difference method has been the most popular of the numerical techniques; however, for structures of composite materials and of arbitrary geometry, this procedure is difficult to apply. In the present investigation, the finite element idealization is used as the basic numerical procedure. This technique has been applied successfully in the stress analysis of many complex structures.13 In Ref. 4 impressive results were obtained in the analysis of axisymmetric shells approximated by a series of truncated cone elements. The approach, which is presented here, is similar in many respects to existing methods used in the analysis of two-dimensional stress problems.57 Recently, the finite element method was applied to the structural analysis of axisymmetric solids subjected to axisymmetric loads.8 In the present paper, the finite element method is used in the determination of stresses and displacements developed within elastic solids of revolution which are subjected to axisymmetric or nonaxisymmetr ic loads. Emphasis is placed on the application of the technique to complex aerospace structures.

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

Real-world groundwater modeling deals with heterogeneous and anisotropic aquifer systems. Numerical models, particularly the finite difference and finite element methods, are extensively used for modeling such a complex aquifer system. The finite difference method is an age-old straightforward approach, whereas the finite element method is more idealistic, relatively new, and has emerged as a popular approach. The unique advantages with the finite element method are (i) an irregular-shaped domain can be discretized into elements with a matching boundary, (ii) a node can be connected with the neighboring nodes in any direction. Thus, the finite element analysis allows water/solute to flow more freely toward the surrounding nodes along nodal links. On the other hand, the mathematical formulation of the finite element technique follows a set of rules step by step. Conventionally, this idealistic approach uses shape functions and integration techniques to develop element matrices, which, in turn, are assembled to form the global matrix. This conventional global matrix was found to be inconsistent with the basic partial differential equation representing the mass balance in a nodal domain as a control volume. As a result, the matrix solution, particularly for nonlinear problems (e.g., vadose zone flow, density dependent flow, and so forth) becomes unstable and computationally burdensome. To overcome the limitations of the conventional finite element method for real-world groundwater modeling, a new approach, called the direct-formulation finite element (DFFE) method, has been developed. In this approach, a two-dimensional model domain is discretized into triangular elements. Element matrices are formed directly based on flow principles and material properties of elements without using shape functions and integration process. The DFFE method is robust. The global matrix is consistent, and the solution is stable and efficient. The results obtained from this approach were found in a good agreement with that obtained from analytical and other widely used models.

An adaptive finite element technique for thermal stress analysis of built-up structures has been developed. A finite element formulation for a triangular membrane element and a new plate bending element, used for modelling such structures under both mechanical and thermal loadings, is presented. The associated finite element matrices have been derived in closed form. The performance of the new plate bending element is evaluated for a plate with temperature gradient through its thickness by comparing the predicted solution with the exact solution. The effectiveness of the adaptive meshing technique combined with the finite element method is evaluated by thermal stress analysis of a built-up structure with intersecting panels. The application demonstrates that the adaptive meshing technique can provide an accurate solution with fewer elements and shorter computational time than the standard finite element procedure.

For certain continuum problems, it is desirable and beneficial to combine two different methods together in order to exploit their advantages while evading their disadvantages. In this paper, a bridging transition algorithm is developed for the combination of the meshfree method (MM) with the finite element method (FEM). In this coupled method, the MM is used in the sub-domain where the MM is required to obtain high accuracy, and the FEM is employed in other sub-domains where FEM is required to improve the computational efficiency. The MM domain and the FEM domain are connected by a transition (bridging) region. A modified variational formulation and the Lagrange multiplier method are used to ensure the compatibility of displacements and their gradients. To improve the computational efficiency and reduce the meshing cost in the transition region, regularly distributed transition particles, which are independent of either the meshfree nodes or the FE nodes, can be inserted into the transition region. The newly developed coupled method is applied to the stress analysis of 2D solids and structures in order to investigate its’ performance and study parameters. Numerical results show that the present coupled method is convergent, accurate and stable. The coupled method has a promising potential for practical applications, because it can take advantages of both the MM and FEM when overcome their shortcomings.

A finite element method (FEM) for solution of the conservative full potential flow equation is presented as a means for improving the computation of transonic flows about complex geometries. The method does not require an orthogonal mesh, thus removing a major constraint in grid generation. A standard Galerkin finite element formulation is used in conjunction with the artificial compressibility method which stabilizes the algorithm in transonic flow and allows the capture of embedded shock waves. Solution comparisons, including one test case which stresses the'capabilities of state-of-the-art finite difference methods, verify the FEM accuracy and geometric capability. I. Introduction W HILE the computation of inviscid two- and threedimensional flowfields by finite difference methods has now become commonplace in industry, one of the most important technological deficiencies is in the area of geometric modeling, particularly for transonic flows. The finite element method (FEM) is presented as a way of alleviating some problems of geometric modeling which limit the effectiveness of current finite difference methods. Certain specific advantages of the FEM, if fully exploited, may improve greatly the computational state-of-the-art. Since the governing equations of inviscid transonic potential flow are nonlinear, the method of superposition (i.e., panel methods) is insufficient to yield a proper solution, even though solutions have been obtained by a method combining surface singularities and field differences.1 Also, because of the nonlinearity, similarity laws do not exist which allow subsonic flows to be related to transonic flows. Field methods must be used, typically finite difference or finite element methods, requiring a grid of points to be defined, both on the body surface arid in the flowfield, on which solutions to the governing equations are approximated. Finite difference method (FDM) computer codes generally

This article provides a detailed accuracy study of various methods that are exposed in textbooks to for approximate evaluation of buckling loads by solving problems with known closed-form solutions. The inverse buckling problem for the inhomogeneous column is chosen with the known exact solution. The approximate methods utilized in the analysis are the Galerkin method, the finite element method (FEM), and the finite difference method (FDM). For each method, multiple boundary conditions for the columns, those being simply supported-simply supported (S-S) and clamped-clamped (C-C), are solved. These solved examples provide ample material for the interested reader to be acquainted with issues of accuracy achieved by each method. Whereas there are theorems for convergence, it is instructive to have actual demonstration of the convergence.

Accurate computational modeling of electric fields in the human head has become important in clinical research to study or influence brain functionality. While existing numerical approaches have been evaluated against simple geometries with known closed form solutions, the relationship between these approaches in more complex geometries has not been studied. Here, we compare the three most commonly used approaches for bioelectric modeling: the finite element method (FEM), the finite difference method (FDM), and the boundary element method (BEM). Using both isotropic and anisotropic conductivity distributions, we construct and compare bioelectric models for a realistic head geometry. Our results suggest that both FEM and FDM are capable of accurately model voltages in the brain, while computations from BEM result in significantly larger errors, due to the increased simplicity and implicit model assumptions.

Comparison of Eulerian and Lagrangian numerical techniques for the Stokes equations in the presence of strongly varying viscosity

A practical procedure to assess critical defects in pressure vessels subjected to fatigue loads

The population density approach has been used for modeling the dynamics of large-scale neuronal networks to consider the stochastic nature of the signal processing in the brain. From the conservation principle, the time evolution of population density can be described by a nonlinear partial differential equation (PDE). The finite difference method (FDM) has been proposed for solving this PDE. However, FDM is sensitive to the density gradient of the solution and is confined to problems that have a regular state space. An irregular state space is always obtained if realistic neuronal models are considered. In this study, the finite element method (FEM) is formulated to solve this POE and apply it to solve the orientation tuning problem. The results show that when the state space discretization is coarse, FEM retains high accuracy whereas FDM does not. Furthermore, FEM reduces computation time by 90% compared to that required for FDM. In addition, FEM can easily handle the existence of a point source without any modifications. Due to its superior accuracy, efficiency, and consistency, FEM may be a better numerical technique for applying the population density approach to large-scale neuronal networks.

Static and dynamic analysis of two-phase peridynamic one-dimensional systems is thoroughly investigated in this paper for various boundary conditions by exact and numerical approaches. Such a nonlocal study concerns the axial or torsional responses of two-phase peridynamic rods or the behavior of a two-phase peridynamic string. For a normalized exponential kernel, the integro-differential equation of the two-phase peridynamic problem can be converted as a linear higher-order differential equation with constant coefficients. Exact static solutions of fixed–fixed and fixed–free two-phase peridynamic axial rod (TPPAR) under uniform distributed loading and concentrated force are analytically derived. It is shown that the pure peridynamic rod problem can be ill-posed due to a conflict between the natural boundary condition and the peridynamic constitutive boundary condition. The two-phase peridynamic problem may be justified from physical arguments, and is also seen as a regularized peridynamic problem. The exact eigen-frequencies of the two-phase peridynamic rods are also analytically obtained for various boundary conditions, including fixed–free, fixed–fixed or free–free boundary conditions. These theoretical approaches are corroborated by numerical investigations based on Finite Difference Method (FDM), Finite Element Method (FEM), exact FEM and variational iterative methods. Two FDM schemes are presented for the statics and the vibration of two-phase peridynamic rods, with convergence orders of 1 and 2, respectively. Moreover, three different FE procedures are proposed as: (1) FEM for direct solving the integro-differential equations of two-phase peridynamic one-dimensional system, (2) FEM for discretizing the pure differential equations, and (3) Exact FEM to obtain both the exact stiffness matrix and exact dynamic stiffness matrix of two-phase peridynamic one-dimensional system, based on Eisenberger’s methodology. The order of convergence for the first and the second proposed FEM is equal to 2 and 4, according to the type of selected finite elements. Moreover, the third proposed FEM based on the exact shape function of the peridynamic problem gives the exact solution from one single FEM element. In addition, the variational iterative method of Stodola–Vianello is successfully applied to the two-phase peridynamic eigenvalue problems.

Different numerical methods including 2-D finite element (FE) analysis were implemented for hypothetical raft foundations of high length/breadth ratios to evaluate the reliability of (1) Boussinesq’s method, (2) “rigid-punch” (RP) elastic methods and (3) finite difference (FD) incorporating subgrade reaction theory. A dedicated computer program was developed and used in addition to professional software packages to analyse uniformly loaded rafts of various length/width ratios and stiffnesses, interacting with multi-layered soils of different properties. It was found that the greater the flexibility of the raft the closer was the agreement between the deformations computed from FE and FD methods and the settlements predicted from FE and RP methods. Additionally, the FE method yielded the most realistic predictions principally because of its capability to model and account for the interactions between soil layers as well as the continuity of the soil support. The results revealed that soil stresses computed from elastic methods were inaccurate in cases where a relatively incompressible bedrock existed within depths equivalent to 4 times the raft width or less. This led to a recommendation that in order to predict raft deformations accurately, only those methods that account for both soil-structure and soil–soil interaction should be used in designing raft foundation in practice.

Arnold Verruijt (1940‐2022)

This paper explores the application of computerized finite element modeling techniques to the design of complex joints such as those used in semisubmersible drilling platforms. The structural complexity of such joints results in complex stress fields and a lack of suitable analytical methods has caused designers to rely heavily on intuition and simple "cook book" formulas. The advent of the computerized finite element technique fills the analytical void, but experience has shown that practical economic considerations apply certain limits here also. The joint analyst seldom has the time or money available to subdivide his finite element mesh until he has achieved convergence to the "exact solution. The purpose of this study was to develop guidelines for finite element mesh layouts which would yield stress field information adequate for the joint designer's needs. Five typical joint designs were modeled by finite elements j three by manually laying out the mesh and two via an automatic mesh generation program recently developed at Southwest Research Institute. These finite element idealizations were analyzed using the computer program STARDYNE. The results are compared with experimental stress analyses of scale models performed at Southwest Research Institute. For the cases studied, it is concluded that the finite element method gives results which are within about 20 percent of the experimental results if the elements near cylinder intersections have an arc angle of about 15 degrees along the intersection line and a length of square root rt/2 normal to the intersection line. INTRODUCTION The design of the complex joints of offshore drilling rigs is a problem frequently encountered for which no simple solution is available. For example, in the case of large semisubmersible and fixed platforms in deep water, the joints are frequently stiffened with rings, gussets, bulkheads, and insert plates. This constitutes a relatively complex structure so that powerful analytical techniques must be used to produce an adequate design. Selective reinforcement of a joint at critical regions is the objective of a typical joint design. Intuition and "cook book" methods play an important role in this process. However, too much reliance on these methods may prove dangerous. In general, most designers are well aware that, without detailed knowledge of the stress distribution in a complex joint, one may tend to stiffen an apparently weak area with a resulting increase in stresses in other areas of the joint. Until recently the only effective means of determining stresses in a complex joint has been to place strain gages on scale models of a proposed structure and locate problem areas by a "cut-and-try" technique, an expensive procedure at best. In the last few years, however, the finite element method, coupled with the digital computer, has provided the structural analyst with a powerful and cost-effective tool to analyze these joints.