Topology optimization with displacement-based nonconforming finite elements for incompressible materials

This investigation is concerned with the topology optimization using displacement-based nonconforming finite elements for problems involving incompressible materials. Although the topology optimization with mixed displacement-pressure elements was performed, a displacement-based approach can be an efficient alternative because it interpolates displacement only. After demonstrating the Poisson locking-free characteristics of the employed nonconforming finite elements by a simple patch test, the developed method is applied to solve the design problems of mounts involving incompressible solid or fluid. The numerical performance of the nonconforming elements in topology optimization was examined also with existing incompressible problems. This paper was recommended for publication in revised form by Associate Editor Tae Hee Lee Gang-Won Jang received his M.S. degree in 2000, and Ph.D. degree in 2004, both from the School of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Korea. He is currently an Assistant Professor at the School of Mechanical and Automotive Engineering, Kunsan National University, Jeonbuk, Korea. His current interest concerns topology optimization of multiphysics problems and thin-walled beam analysis. Yoon Young Kim received his B.S. and M.S degrees from Seoul National University, Seoul, Korea, and the Ph.D. degree from Stanford University, Palo Alto, CA, in 1989. He has been on the faculty of the School of Mechanical and Aerospace Engineering, Seoul National University, since 1991. He is also the Director of the National Creative Research Initiatives Center for Multiscale Design. His main research field is the optimal design of multiphysics systems, mechanisms, and transducers. He has served as an editor of several Korean and international journals, and as an organizing committee member of several international conferences.     

On the linear elasticity of porous materials

In this work, a porous material is represented as a composite made of an isotropic matrix and spheroidal inclusions with zero stiffness, describing the pores. The shape of the pores ranges from flat to spherical to fibre-like, and their orientation is assumed to obey a given probability distribution function. As an example of straightforward physical interpretation, the isotropic case of randomly oriented voids-pores is studied as a function of the matrix Poisson's ratio and the porosity (void volumetric fraction). The results are in good agreement with published experimental data on porous metals, and with previous isotropic solutions in the literature.     

An exact finite strip for the calculation of relative post-buckling stiffness of I-section struts

This paper presents the theoretical developments of an exact finite strip for the buckling and initial post-buckling analyses of I-section struts. The so-called exact strip is developed based on the concept that it is effectively a plate. The presented method, which is designated by the name Full-analytical Finite Strip Method, provides an efficient and extremely accurate buckling solution. In the development process, the Von-Karman's equilibrium equation is solved exactly to obtain the buckling loads and mode shapes for the I-section struts. The investigation of buckling behavior is then extended to an initial post-buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. The current post-buckling study is effectively a single-term analysis, which is attempted by utilizing the so-called semi-energy method. Through the solution of the Von-Karman's compatibility equation, the in-plane displacement functions which are themselves related to the Airy stress function are developed in terms of the unknown coefficient in the assumed out-of-plane deflection function. All the displacement functions are then substituted in the total strain energy expressions. The theorem of minimum total potential energy is subsequently applied to solve for the unknown coefficient. Finally, the developed method is subsequently applied to analyze the initial post-buckling behavior of some representative I-sections for which the results were also obtained through the application of a Semi-energy Finite Strip Method [Ovesy HR. The development and application of a semi-energy post-local buckling finite strip. PhD dissertation, Cranfield University, UK, 1998]. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is significantly promoted.     

Reverse problem of the theory of elasticity for drawing die supported by a cage

A theoretical analysis for the determination of the interference function in order to press in a reinforcing cage of carbide die, which provides a decrease in the stress concentration on the internal profile of die, was performed on the basis of a model of a rough friction surface and the simultaneous failure mode approach. A closed system of algebraic equations was designed, which enables to solve the problem of optimal projection of carbide die depending on the geometrical and mechanical characteristics of the die and cage. The press-in interference determined provides an increase in the load-carrying capability of carbide die supported by a cage.     

Mean dilatational based finite element formulation for a nearly incompressible nonlinear large deformation analysis

The finite element method (FEM) is one of the most powerful tools that can be used to solve engineering problems. Standard displacement-based isoparametric elements are the most popular in the engineering community for their simple implementation. However, low-order displacement-based elements exhibit locking in the incompressible. Our proposed formulation is a locking removing method that is free from the need to store any extra integration point. We calculate the modified deformation gradient from the mean dilatation of the element and also modify the finite element equations using the mean dilatation. This method is also different from the B-bar method as the B matrix is not modified, but rather the weak form is linearized for the modified deformation gradient. Therefore, this method is a unique implementation of the average dilatation of the element. Here we also present an implementation algorithm of the proposed formulation and compare the results with available methods.

     

A new beam finite element for the analysis of functionally graded materials

A new beam element is developed to study the thermoelastic behavior of functionally graded beam structures. The element is based on the first-order shear deformation theory and it accounts for varying elastic and thermal properties along its thickness. The exact solution of static part of the governing differential equations is used to construct interpolating polynomials for the element formulation. Consequently, the stiffness matrix has super-convergent property and the element is free of shear locking. Both exponential and power-law variations of material property distribution are used to examine different stress variations. Static, free vibration and wave propagation problems are considered to highlight the behavioral difference of functionally graded material beam with pure metal or pure ceramic beams.     

Some relations for determining the wear of composite brake materials

Investigation of several types of composite brake materials showed that both the mean temperature of the friction surface and the wear rate of the materials vary non-linearly as a function of load and speed. The presence of some metal inclusions leads to a less complex non-linear relationship of the wear rate.     

Inverse doubly periodic problem of elasticity theory for a composite reinforced by unidirectional fibers

The problem of how to determine the optimal form of a fiber cross section to decrease the concentration of stresses in the composite has been solved by using the uniform-strength principle. The fiber cross section form determined increases the load-carrying capability of the composite solid.     

Influence of different electrical boundary conditions on the elasticity solutions of piezoelectric plane beam

Influence of different electrical boundary conditions (BCs) on the elasticity solutions of piezoelectric plane beam (PPB) is investigated using analytical technique. The first case is considering electrical displacement of the two longitudinal sides of PPB. The second case is electrical potential. Firstly, the unified equations to obtain the elasticity solutions of PPB corresponding to these two cases are given briefly. Secondly, two examples are given to verify the correctness of the theoretical formulations presented in this paper. Finally, the responses of PPB acted by the same mechanical loads but with different electrical BCs are compared.     

黑板原理在级进模工步排样方案构造系统中的应用研究

首先对黑板原理及相关技术进行了简单阐述。全文着重介绍了构造复杂级进模工步排样方案生成系统中，黑板模型的建立，并论述了该模型的推理机制。随后，还介绍了黑板结构的实现，通过原型系统的开发，论证了该理论在系统实现中的合理性和优越性。     

Three-dimensional incompressible flow of a gas through feed holes into an annular space of finite length

This paper presents a theoretical model of the three-dimensional flow of an incompressible gas in a concentric double-plane admission externally pressurised gas journal bearing. Inertia effects and the discreteness of the feed holes are included. The flow map demonstrates the formation of a vena contracta at the commencement of the film, downstream of the feed holes, as well as detailed behaviour of the streaklines within the film. The pressure profile obtained shows a depression at the vena contracta, followed by a partial recovery which varies around the feed holes, depending on whether the local flow is exiting between the rows of holes or outboard towards atmosphere. A comparison is also made with a similar one-dimensional incompressible flow solution to obtain flow and pressure factors, to account for the discreteness of the feed holes as well as inertia effects.     

The unified equations to obtain the exact solutions of piezoelectric plane beam subjected to arbitrary loads

The unified equations to obtain the exact solutions for piezoelectric plane beam subjected to arbitrary mechanical and electrical loads with various ends supported conditions is founded by solving functional equations. Comparing this general method with traditional trial-and-error method, the most advantage is it can obtain the exact solutions directly and does not need to guess and modify the form of stress function or electric displacement function repeatedly. Firstly, the governing equation for piezoelectric plane beam is derived. The general solution for the governing equation is expressed by six unknown functions. Secondly, in terms of boundary conditions of the two longitudinal sides of the beam, six functional equations are yielded. These equations are simplified to derive the unified equations to solve the boundary value problems of piezoelectric plane beam. Finally, several examples show the correctness and generalization of this method.     

Some aspects of optimizing contrasts for the investigation of joint materials in the environmental scanning electron microscope

In the environmental scanning electron microscope, material joints of different atomic mass and different electrical conducting properties can easily be observed simultaneously without coating the specimen. For such heterogeneous materials, the quality of the image can be optimized with respect to contrast and resolution if the contrast types as well as their significance to the composition of the image are known.     

A combined contact elasticity and finite element-based model for contact load and pressure distribution calculation in a frictional workpiece-fixture system

Contact forces between workpiece and fixture define fixture stability during clamping and influence workpiece accuracy during machining. In particular, forces acting in the contact region are important for understanding deformation of the workpiece at the contact region. This paper presents a model that combines contact elasticity with finite element methods to predict the contact load and pressure distribution at the contact region in a workpiece-fixture system. The objective is to determine how much clamp forces can be applied to generate adequate contact forces to keep the workpiece in position during machining. The model is able to predict the normal and tangential contact forces as well as the pressure distribution at each workpiece-fixture contact in the fixturing system. Model prediction is shown to be in good agreement with known industry practice on clamp force determination. The presented method has no limits on the types of materials that can be analyzed.     

Some numerical slipline field solutions for drawing through circular dies

A method is presented for the generation of slipline fields and their hodographs for the problem of plane strain drawing a rigid, perfectly plastic (rpp) material through frictionless circular dies. An optimization scheme is given, using a readily available multivariable optimization routine, which will reliably adjust the shape of an assumed slipline to ensure hodograph convergence. A number of programme runs using different geometries show some interesting oscillatory variations in the computed stress distributions along the die surface and across the exit slipline.     

Some improvements on the unfolding inverse finite element method for simulation of deep drawing process

The linear unfolding inverse finite element method (IFEM) has been modified and enhanced by implementing large deformation relations. The method is helpful to predict forming severity of the part that should be deep drawn as well as its blank shape and strain distribution in preliminary design stage. The approach deals with minimization of potential energy and large deformation relations with membrane elements. To reduce the computation time, the part is unfolded properly on the flat sheet and treated as 2D problem. Moreover, the nonlinear stress-strain relationship of plastic material properties has been considered to increase the accuracy of the results. An experiment set up has been prepared to form a rectangular cup. Then, the obtained cup has been analyzed by linear unfolding IFEM and the proposed method. Comparisons of the measured thickness strains and the blank shape show that the proposed method predicts the strain distribution more accurately than the linear method.     

Inverse theory of elasticity problem of mounting a disk on a rotating shaft

A theoretical analysis of the determination of the tightness of mounting an elastic disk on a rotating shaft ensuring the absence of stress concentration on the inner rim of the disk, i.e., constant normal circular (optimal) stress, is performed on the basis of the equal-strength principle. The value of optimal constant stress is determined upon solution of the optimization problem. A closed system of algebraic equations providing the possibility of solving the optimal design problem is constructed. The found mounting tightness ensures an increase in disk bearing capacity.     

Modified membrane finite element formulation for sheet metal forming analysis of planar anisotropic materials

A finite element formulation is derived for sheet metal forming analysis of planar anisotropic materials. The formulation incorporates membrane elements whereas it takes the bending effect into account explicitly. The strain energy term in the formulation is decomposed into the membrane energy term for mean stretching and the bending energy term for pure bending. This procedure needs careful evaluation for the orientation of the anisotropic axes. The formulation is then combined with an effective algorithm to calculate distribution of the blank holding force in each step according to the thickness in the flange region. The calculation employs a special relation between the thickness and the blank holding force. The simulation examples demonstrate the validity and versatility of the developed computer code by showing that the thickness variation in the flange region redistributes the blank holding force during the deformation. The present algorithm can predict accurate deformed shapes and thickness strain distribution with the anisotropy of materials and the variable blank holding force.