Fixed Grid Finite Element Method Research Articles

Topological design of 3D phononic crystals for ultra-wide omnidirectional bandgaps

As a unique kind of artificial periodic composite materials, phononic crystals have attracted the wide attention in recent years. Unremitting efforts have been made to investigate the potential existence of phononic bandgaps, which can be used to manipulate the propagation of acoustic/elastic waves. This paper proposes a new topology optimization algorithm to achieve the optimised bandgap structures of 3D phononic crystals based on fixed-grid finite element method (FEM) and evolutionary procedure. The maximum normalised gap ratio obtained is as large as 97% between the 12th and 13th order of bands, with the optimised structure of tungsten carbide spheres embraced in epoxy matrix. Symmetric unit cells for simple cubic (SC) lattice are presented for the specified bandgaps to validate the effectiveness of the proposed optimization algorithm for designing 3D phononic bandgap crystals. Meanwhile, the proposed topology optimization represents structures with clear topologies and smooth boundaries.

Topology optimization for heat transfer enhancement in Latent Heat Thermal Energy Storage

Performance of a Latent Heat Thermal Energy Storage depends strongly on the spatial layout of high conductive material and phase change material. Previous design studies have explored a limited design space and have rarely taken advantage of any formal optimization approach. This paper presents a topology optimization framework of a Thermal Energy Storage system involving phase change. We solve the Stefan problem for solidification with a fixed grid finite element method based on the apparent heat capacity technique, while the topology optimization problem is formulated using a density-based method. This approach allows to identify design trends that have been rarely investigated in the past. Firstly, we explore the inherent trade-off between discharged energy and required time for complete discharge. We obtain very different designs and highly varying performances at selected Pareto points. Secondly, by comparing results obtained in two and three dimensions we observe that 3D designs allow superior performances by presenting features that are not apparent in 2D. Thirdly, we propose a formulation of the design problem that yields a nearly constant thermal power output during the entire discharge process. If the maximum discharge time is sufficiently large, the optimized design presents fins that are disconnected from the internal tube.

OPTIMAL SHAPE DESIGN FOR HEAT CONDUCTION USING SMOOTHED FIXED GRID FINITE ELEMENT METHOD AND MODIFIED FIREFLY ALGORITHM

The present study is concerned with optimal shape determination of inhomogeneous and temperature dependent domains under steady state heat conduction. Such situations are important in many thermal design problems, especially in shape design of electronic components and chips. In the present paper, we formulate the shape optimization problem based on volume minimization of heat conductive material while limiting maximum temperature. The smoothed fixed grid finite element method which is a new approach based on the non-boundary-fitted meshes is used to obtain temperature field. The boundary parameterization technique using splines is also adopted to manipulate the shape variations. A modified version of the firefly algorithm which is a recently developed metaheuristic optimization technique is proposed as the optimizer. These modifications consist of adding memory, adding newborn fireflies and proposing a new updating formula. To evaluate the applicability of the proposed method five numerical examples are solved and the results are presented.

Numerical flow simulation in gated hydraulic structures using smoothed fixed grid finite element method

Successful design and operation of hydraulic structures require more effective and reliable tools to be used in a variety of practical problems, including bottom outlets, intakes and/or spillways. Until recently, physical modeling has been the principal approach in studying the flow pattern and behavior of such structures. The main concerns might generally be related to estimating of discharge coefficients, frictional losses, details of local flow patterns, position of the free surface, and air entrainment. The application of a new developed numerical modeling for free surface flow simulation in gated tunnels is presented here. Among various parameters, the free surface profile, pressure distribution and discharge of the flow passing through gated hydraulic structures are considered in the present study. The solution of free surface flow problems is usually carried out through an iterative process due to unknown geometry and nonlinear boundary conditions. Therefore, the solution of these problems using traditional numerical methods presents some difficulties since the computational mesh needs to be modified for each iteration. The application of the smoothed fixed grid finite element method in the solution of free surface flow problems is investigated in this paper. The main advantage of the proposed method is that it is based on non-boundary-fitted meshes which simplify the solution procedure. In this method, the gradient smoothing technique is used to facilitate formulation of boundary intersecting elements. To evaluate the applicability of the proposed method, three representative examples are solved and the results are compared with experimental and numerical results available in the literature. The results of the present study indicated the power of smoothed fixed grid finite element method in solving the free surface potential flows through gated hydraulic structures.

Approximation of the effective moduli of particulate composite with the fixed grid finite element method

The fixed grid finite element method is introduced to approximate the effective moduli of particulate composite. The difficulty of domain discretization induced by the inclusions is avoided in the fixed grid finite element method, as the discretization is independent of the inclusions. The elastic properties of every finite element are approximated by a weighted function, and the volume fractions of the constituents in the element are taken as the weights. A simple scheme is proposed to approximate the volume fractions in every element. The validities of the fixed grid finite element method are verified, by comparing the effective moduli obtained from the fixed grid finite element method with those obtained from finite element method. The anisotropy of particulate composite is discussed with the fixed grid finite element, and the applicability of the fixed grid finite element method for composite with multiple irregular inclusions is illustrated.

Three dimensional smoothed fixed grid finite element method for the solution of unconfined seepage problems

A three dimensional numerical analysis for unconfined seepage problems in inhomogeneous and anisotropic domains with arbitrary geometry is presented in this paper. The unconfined seepage problems are nonlinear in its nature due to unknown location of the phreatic surface and nonlinear boundary conditions which complicates its solution. The presented method is based on the application of non-boundary-fitted meshes and is an extension of the recently proposed two dimensional smoothed fixed grid finite element method. The main objective of using this method is to facilitate solution of variable domain problems and improve the accuracy of the formulation of the boundary intersecting elements. In this method, the gradient smoothing technique is used to obtain the element matrices. This technique simplifies the solution significantly by reducing the volume integrals over the elements into area integrals on the faces of smoothing cells. To locate the free surface, an initial guess for the unknown geometry is selected and modified in each iteration to eventually satisfy nonlinear boundary condition. The application of the proposed technique for three dimensional seepage problems is carried out for different examples including rectangular, trapezoidal and semi-cylindrical dams and the results are compared with those available in the literature.

Inverse geometry heat conduction analysis of functionally graded materials using smoothed fixed grid finite elements

A shape identification scheme is developed in this article to determine the shape of the inaccessible parts of a two-dimensional object made of functionally graded material using the measured temperatures on its accessible parts. The proposed method is based on minimization of the differences between measured and calculated temperatures. The smoothed fixed grid finite element method which is a new approach based on the non-boundary-fitted meshes and the gradient smoothing technique are used for the solution of direct problem and shape sensitivity analysis. The boundary parameterization technique using splines is also adopted to manipulate the shape variations. The conjugate-gradient method is used as the optimization algorithm and some numerical examples are solved to evaluate the applicability of the proposed method in the context of functionally graded materials.

Compliant mechanisms design based on pairs of curves

The success of compliant mechanism design by structural topology optimization approach depends, to a large extent, on its structural geometry representation scheme. In this work, a novel representation scheme based on pairs of curves is presented. In the representation, the structure is characterized by a set of input/output (I/O) regions. While it is still unknown how the rest of the design space will be occupied by the structure, the I/O regions must exist somewhere because any structure must have parts which interact with its surroundings by way of at least one loading region, one support region, and one output region. For a valid structural design, pairs of Bezier curves are used to connect I/O regions in order to form one single connected load-bearing structure. The boundary is explicitly described, so the need for smoothening of the blurred and jagged edges can be avoided by developing such a representation scheme to directly generate smooth boundary structures. With the scheme, shape and topology can be optimized simultaneously, and the obtained topology solutions have no check-board phenomena nor intermediate zones. A multi-objective genetic algorithm is then applied to couple with the representation scheme for defining and encoding the structural geometry in the form of graph. The solution framework is integrated with a nonlinear fixed grid finite element method (FG-FEM) code for large-displacement analyses of the compliant structures. Simulation results from a displacement inverter indicated that the proposed representation scheme is appropriate.

Unconfined seepage analysis in earth dams using smoothed fixed grid finite element method

SUMMARYOne major difficulty in seepage analyses is finding the position of phreatic surface which is unknown at the beginning of solution and must be determined in an iterative process. The objective of the present study is to develop a novel non‐boundary‐fitted mesh finite‐element method capable of solving the unconfined seepage problem in domains with arbitrary geometry and continuously varied permeability. A new non‐boundary‐fitted finite element method named as smoothed fixed grid finite element method (SFGFEM) is used to simplify the solution of variable domain problem of unconfined seepage. The gradient smoothing technique, in which the area integrals are transformed into the line integrals around edges of smoothing cells, is used to obtain the element matrices. The solution process starts with an initial guess for the unknown boundary and SFGFEM is used to approximate the field variable. The boundary shape is then modified to eventually satisfy nonlinear boundary condition in an iterative process. Some numerical examples are solved to evaluate the applicability of the proposed method and the results are compared with those available in the literature. Copyright © 2011 John Wiley & Sons, Ltd.

Static and dynamic analysis of 2D and 3D elastic solids using the modified FGFEM

In this paper, a modification on the fixed grid finite element method (FGFEM) is presented and used for the static and dynamic analysis of 2D and 3D elastic solids. In this study, fixed non-boundary-fitted meshes are used to solve elasticity problems. A new approach for computing stiffness matrix of boundary intersecting elements is presented. Natural boundary conditions applied via numerical integration of traction forces and essential boundary conditions applied using a technique based on penalty function on the non-conforming boundaries. The proposed method is applied to some 2D and 3D problems in linear elasticity and the results show good agreement with analytic solutions. This method is also used for the natural frequency analysis of elastic solids and the results show good agreement with those obtained using the finite element method.

Solution of geometric inverse heat conduction problems by smoothed fixed grid finite element method

This paper presents a methodology for solving the nonlinear inverse geometry heat transfer problems where the observations are temperatures measured at points on the external boundary and the unknowns are the shape and the location of a cavity inside the problem domain. The algorithm is based on the minimization of the squared error between the measured temperatures and those calculated by the proposed smoothed fixed grid finite element method where a non-boundary-fitted mesh is used to solve the direct problem. This eliminates mesh adaptation and re-meshing processes as needed in the standard finite element method and reduces the analysis cost significantly. The proposed method uses the advantages of smoothed finite element method and fixed grid finite element method in which a fixed computational mesh is used to solve the variable domain problem and the numerical integration is performed through gradient smoothing technique. The present method is more accurate and facilitates integration over boundary intersecting elements. The shape gradients of the shape identification problem are obtained by direct differentiation method. The domain parameterization technique with cubic splines is adopted to manipulate the shape variation of the cavity. To stabilize the optimization process, a geometric regularization function is used. Some numerical examples are solved to evaluate the method. In these examples effects of initial guess, measurement errors, cavity shape and cavity size on the results are examined. It is shown that the results are in good agreement with exact solutions.

Structural optimization of as-built parts using reverse engineering and evolution strategies

In industry, some parts are prone to failures or their design is simply sub-optimal. In those critical situations, one would like to be able to make changes to the part, making it lighter or improving its mechanical resistance. The problem of as-built parts is that the original computer-aided design (CAD) model is not available or is lost. To optimize them, a reverse engineering process is necessary to capture the shape and topology of the original design. This paper describes how to capture the original design geometry using a semi-automated reverse engineering process based on measurement provided by an optical 3D sensor. Following this reverse engineering process, a Fixed Grid Finite Element method and evolutionary algorithms are used to find the optimum shape that will minimize stress and weight. Several examples of industrial parts are presented. These examples show the advantages and disadvantages of the proposed method in an industrial scenario.

Extension of the fixed grid finite element method to eigenvalue problems

This paper details the development of the fixed grid FEA (FGFEA) method to the solution of eigenvalue (natural frequency and buckling) problems, using a 4 node 20 degree of freedom shell element. The formulation of the FG element stiffness, buckling and natural frequency equations is presented. The first natural frequency and buckling load are determined for the simply supported plate with a central circular void. The void is then offset to three further locations to asses the applicability of FGFEA eigenvalue analysis at reduced symmetry. A correlation trend is formed by comparison with traditional FEA results at nine increasingly refined mesh densities for each example. In addition to this, a relationship between the FGFEA error and the boundary element intersection angle with the void is formed.

A strength-based multiple cutout optimization in composite plates using fixed grid finite element method

The design of interior cutouts in laminated composite panels is of great importance in aerospace, automobile and structural engineering. Based on the Tsai–Hill failure criterion of the first ply, this paper presents a newly developed Fixed (FG) Grid Evolutionary Structural Optimization (ESO) method to explore shape optimization of multiple cutouts in composite structures. Different design cases with varying number of cutouts, ply orientations and lay-up configurations are taken into account in this study. The examples demonstrate that the optimal boundaries produced by FG ESO are much smoother than those by traditional ESO. The results show the remarkable effects of different opening numbers and various lay-up configurations on resulting optimal shapes. The paper also provides an in-depth observation in the interactive influence of the adjacent cutouts on the optimal shapes.

Shape optimisation of continuum structures via evolution strategies and fixed grid finite element analysis

Evolution strategies (ES) are very robust and general techniques for finding global optima in optimisation problems. As with all evolutionary algorithms, ES apply evolutionary operators and select the most fit from a set of possible solutions. Unlike genetic algorithms, ES do not use binary coding of individuals, working instead with real variables. Many recent studies have applied evolutionary algorithms to structural problems, particularly the optimisation of trusses. This paper focuses on shape optimisation of continuum structures via ES. Stress analysis is accomplished by using the fixed grid finite element method, which reduces the computing time while keeping track of the boundary representation of the structure. This boundary is represented by b-spline functions, circles, and polylines, whose control points constitute the parameters that govern the shape of the structure. Evolutionary operations are applied to each set of variables until a global optimum is reached. Several numerical examples are presented to illustrate the performance of the method. Finally, structures with multiple load cases are considered along with examples illustrating the results obtained.

Modeling of oxidation and its effect on crack growth in titanium alloys

The effect of oxidation on the crack growth resistance of titanium alloys is investigated in this work. The oxidation process is modeled by modifying the Fickian diffusion problem in order to account for the chemical reaction (phase change) in the material. Two different variants of a fixed grid finite element method for numerical simulation of oxidation are used. The first approach taken is to locate the oxidation front and split the domain into metal and oxide subdomains. The second approach is based on reformulating the diffusion equations in both the oxide and metal, resulting in a single non-linear equation for the whole domain. After the oxidation process is modeled, the mechanical analysis is performed. In the finite element formulation of the mechanical problem, the influence of oxidation is introduced by the change of the elastic constants and the oxidation-induced eigenstrain. The applicability of both the oxidation and mechanical models is tested by simulating oxidation of a pre-cracked Ti-15-3 compact tension specimen which is then subjected to monotonic loading.

Fixed grid finite elements in elasticity problems

A fixed grid representation of the finite element domain is used to solve elasticity problems. An integration between CAD and FEM systems is made by using this fixed grid representation. Some considerations about FG approximation and stiffness matrix generation are presented. The element stiffness matrix for an element in the mesh is obtained as a factor of a standard element stiffness matrix. A least squares local approximation of stress field is used to calculate the stress at boundary elements. A problem with an analytical solution is used to measure the displacement and stress error. Several meshes and several geometrical configurations of the problem are used to test the reliability of the fixed grid finite element method.

A fixed grid finite element method for nonlinear diffusion problems with moving boundaries

A comprehensive formulation for a class of diffusion problems with non-linear conductivities is derived by unifying and combining the freezing index and Kirchhoff transformation concepts. The transformed equations have appropriate continuity characteristics across the unknown moving boundary. The applicability of the fixed grid algorithm for the total solution domain is, accordingly, demonstrated. Associated finite element formulations and solution procedures for the transformed equations are detailed. In addition, selected numerical results for single and two phase Stefan type problems as well as fluid flow in a prescribed cavity are presented for solution verification and illustration.