Design of Compliant Structural Mechanisms Using B-Spline Elements

2011 ◽  
Author(s):  
Ashok V. Kumar ◽  
Anand Parthasarathy
Keyword(s):  
Inverse Problem ◽  
Topology Optimization ◽  
Compliant Mechanisms ◽  
Spline Approximation ◽  
Structural Response ◽  
Field Application ◽  
Optimization Approach ◽  
Objective Functions ◽  
B Spline ◽  
Spline Basis

Structural design is an inverse problem where the geometry that fits a specific design objective is found iteratively through repeated analysis or forward problem solving. In the case of compliant structures, the goal is to design the structure for a particular desired structural response that mimics traditional mechanisms and linkages. It is possible to state the inverse problem in many different ways depending on the choice of objective functions used and the method used to represent the shape. In this paper, some of the objective functions that have been used in the past, for the topology optimization approach to designing compliant mechanisms are compared and discussed. Topology optimization using traditional finite elements often do not yield well-defined smooth boundaries. The computed optimal material distributions have shape irregularities unless special techniques are used to suppress them. In this paper, shape is represented as the contours or level sets of a characteristic function that is defined using B-spline approximation to ensure that the contours, which represent the boundaries, are smooth. The analysis is also performed using B-spline elements which use B-spline basis functions to represent the displacement field. Application of this approach to design a few simple mechanisms is presented.

Generation and Modification of Non-Uniform B-Spline Surface Approximations to PDE Surfaces Using the Finite Element Method

1990 ◽  
Author(s):  
Joanna M. Brown ◽  
Malcolm I. G. Bloor ◽  
M. Susan Bloor ◽  
Michael J. Wilson
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Spline Approximation ◽  
The Finite Element Method ◽  
B Spline ◽  
B Splines ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Spline Basis ◽  
Element Method

Abstract A PDE surface is generated by solving partial differential equations subject to boundary conditions. To obtain an approximation of the PDE surface in the form of a B-spline surface the finite element method, with the basis formed from B-spline basis functions, can be used to solve the equations. The procedure is simplest when uniform B-splines are used, but it is also feasible, and in some cases desirable, to use non-uniform B-splines. It will also be shown that it is possible, if required, to modify the non-uniform B-spline approximation in a variety of ways, using the properties of B-spline surfaces.

scholarly journals Topology Optimization considering Nonsmooth Structural Boundaries in the Intersection Areas of the Components

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Ruichao Lian ◽  
Shikai Jing ◽  
Zefang Shi ◽  
Zhijun He ◽  
Guohua Song
Keyword(s):  
Topology Optimization ◽  
Geometric Modeling ◽  
Compliant Mechanisms ◽  
Optimization Approach ◽  
Structural Topology ◽  
Topology Description Function ◽  
Structural Responses ◽  
Structure Boundary ◽  
Compliance Problem ◽  
The Impact

In the structural topology optimization approaches, the Moving Morphable Component (MMC) is a new method to obtain the optimized structural topologies by optimizing shapes, sizes, and locations of components. However, the optimized structure boundary usually generates local nonsmooth areas due to incomplete connection between components. In the present paper, a topology optimization approach considering nonsmooth structural boundaries in the intersection areas of the components based on the MMC is proposed. The variability of components’ shape can be obtained by constructing the topology description function (TDF) with multiple thickness and length variables. The shape of components can be modified according to the structural responses during the optimization process, and the relatively smooth structural boundaries are generated in the intersection areas of the components. To reduce the impact of the initial layout on the rate of convergence, this method is implemented in a hierarchical variable calling strategy. Compared with the original MMC method, the advantage of the proposed approach is that the smoothness of the structural boundaries can be effectively improved and the geometric modeling ability can be enhanced in a concise way. The effectiveness of the proposed method is demonstrated for topology optimization of the minimum compliance problem and compliant mechanisms.

Multiobjective Topology Optimization of Compliant Microgripper With Geometrically Nonlinearity

2007 ◽  
Author(s):  
Zhaokun Li ◽  
Xianmin Zhang
Keyword(s):  
Topology Optimization ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Structural Response ◽  
Decision Function ◽  
Element Formulation ◽  
Geometrically Nonlinear ◽  
Conflicting Objectives ◽  
Incremental Scheme ◽  
Moving Asymptotes

Since compliant mechanism is usually required to perform in more than one environment, the ability to consider multiple objectives has to be included within the framework of topology optimization. And the topology optimization of micro-compliant mechanisms is actually a geometrically nonlinear problem. This paper deals with multiobjective topology optimization of micro-compliant mechanisms undergoing large deformation. The objective function is defined by the minimum compliance and maximum geometric advantage to design a mechanism which meets both stiffness and flexibility requirements. The weighted sum of conflicting objectives resulting from the norm method is used to generate the optimal compromise solutions, and the decision function is set to select the preferred solution. Geometrically nonlinear structural response is calculated using a Total-Lagrange finite element formulation and the equilibrium is found using an incremental scheme combined with Newton-Raphson iterations. The solid isotropic material with penalization approach is used in design of compliant mechanisms. The sensitivities of the objective functions are found with the adjoint method and the optimization problem is solved using the Method of Moving Asymptotes. These methods are further investigated and realized with the numerical example of compliant microgripper, which is simulated to show the availability of this approach proposed in this paper.

B-Spline Based Robust Topology Optimization

2015 ◽  
Author(s):  
Yu Gu ◽  
Xiaoping Qian
Keyword(s):  
Topology Optimization ◽  
Optimization Problems ◽  
Compliant Mechanism ◽  
Analytical Description ◽  
Minimal Length ◽  
Optimization Approach ◽  
Material Density ◽  
B Spline ◽  
B Splines ◽  
Robust Formulation

In this paper, we present an extension of the B-spline based density representation to a robust formulation of topology optimization. In our B-spline based topology optimization approach, we use separate representations for material density distribution and analysis. B-splines are used as a representation of density and the usual finite elements are used for analysis. The density undergoes a Heaviside projection to reduce the grayness in the optimized structures. To ensure minimal length control so the resulting designs are robust with respect to manufacturing imprecision, we adopt a three-structure formulation during the optimization. That is, dilated, intermediate and eroded designs are used in the optimization formulation. We give an analytical description of minimal length of features in optimized designs. Numerical examples have been implemented on three common topology optimization problems: minimal compliance, heat conduction and compliant mechanism. They demonstrate that the proposed approach is effective in generating designs with crisp black/white transition and is accurate in minimal length control.

A Bernstein-Schoenberg Type Operator: Shape Preserving and Limiting Behaviour

1995 ◽  
Vol 47 (5) ◽  
pp. 959-973 ◽  
Author(s):  
T. N. T. Goodman ◽  
A. Sharma
Keyword(s):  
Error Estimate ◽  
Spline Approximation ◽  
Type Operator ◽  
Approximation Operator ◽  
Shape Preserving ◽  
B Spline ◽  
Asymptotic Error ◽  
Spline Basis ◽  
Limiting Behaviour ◽  
Shape Preserving Properties

AbstractUsing a new B-spline basis due to Dahmen, Micchelli and Seidel, we construct a univariate spline approximation operator of Bernstein-Schoenberg type. We show that it shares all the shape preserving properties of the usual Bernstein-Schoenberg operator and we derive a Voronovskaya type asymptotic error estimate.

Meshless Local B-Spline Basis Functions-FD Method and its Application for Heat Conduction Problem with Spatially Varying Heat Generation

2013 ◽  
Vol 465-466 ◽  
pp. 490-495 ◽  
Author(s):  
Mas Irfan P. Hidayat ◽  
Bambang Ari-Wahjoedi ◽  
Parman Setyamartana ◽  
Puteri S.M. Megat Yusoff ◽  
T.V.V.L.N. Rao
Keyword(s):  
Heat Conduction ◽  
Heat Generation ◽  
Complex Geometry ◽  
Heat Conduction Problem ◽  
Spline Approximation ◽  
Basis Functions ◽  
Spline Collocation ◽  
B Spline ◽  
Spline Basis ◽  
Spatially Varying

In this paper, a new meshless local B-spline basis functions-finite difference (FD) method is presented for two-dimensional heat conduction problem with spatially varying heat generation. In the method, governing equations are discretized by B-spline approximation in the spirit of FD technique using local B-spline collocation. The key aspect of the method is that any derivative at a point or node is stated as neighbouring nodal values based on the B-spline interpolants. Compared with mesh-based method such as FEM the method is simple and efficient to program. In addition, as the method poses the Kronecker delta property, the imposition of boundary conditions is also easy and straightforward. Moreover, it poses no difficulties in dealing with arbitrary complex domains. Heat conduction problem in complex geometry is presented to demonstrate the accuracy and efficiency of the present method.

Conceptual Design of Structures Using an Upper Bound of von Mises Stress

2018 ◽  
Vol 19 (1) ◽  
Author(s):  
Ashok V. Kumar
Keyword(s):  
Topology Optimization ◽  
Stress Concentration ◽  
Upper Bound ◽  
Von Mises Stress ◽  
Optimization Approach ◽  
Objective Functions ◽  
Stress Concentrations ◽  
Handling Stress ◽  
Design Concepts ◽  
Von Mises

Optimal layouts for structural design have been generated using topology optimization approach with a wide variety of objectives and constraints. Minimization of compliance is the most common objective but the resultant structures often have stress concentrations. Two new objective functions, constructed using an upper bound of von Mises stress, are presented here for computing design concepts that avoid stress concentration. The first objective function can be used to minimize mass while ensuring that the design is conservative and avoids stress concentrations. The second objective can be used to tradeoff between maximizing stiffness versus minimizing the maximum stress to avoid stress concentration. The use of the upper bound of von Mises stress is shown to avoid singularity problems associated with stress-based topology optimization. A penalty approach is used for eliminating stress concentration and stress limit violations which ensures conservative designs while avoiding the need for special algorithms for handling stress localization. In this work, shape and topology are represented using a density function with the density interpolated piecewise over the elements to obtain a continuous density field. A few widely used examples are utilized to study these objective functions.

Reproducing Existing Nacelle Geometries With the Free-Form Deformation Parametrization

2018 ◽  
Author(s):  
Konstantin Rusch ◽  
Martin Siggel ◽  
Richard-Gregor Becker
Keyword(s):  
Inverse Problem ◽  
Spline Approximation ◽  
Aircraft Design ◽  
Free Form ◽  
Design Phase ◽  
B Spline ◽  
Preliminary Aircraft Design ◽  
Least Squares Fit ◽  
Free Form Deformation ◽  
Grid Points

In the conceptual and preliminary aircraft design phase the Free-Form Deformation (FFD) is one of various parametrization schemes to define the geometry of an engine’s nacelle. This paper presents a method that is able to create a C2 continuous periodic approximation of existing reference nacelles with the B-spline based FFD, which is a generalization of the classical FFD. The basic principle of this method is to start with a rotational symmetric B-spline approximation of the reference nacelle, which is subsequently deformed with a FFD grid that is placed around the initial geometry. A method is derived that computes the displacement of the FFD grid points, such that the deformed nacelle approximates the reference nacelle with minimal deviations. As this turns out to be a linear inverse problem, it can be solved with a linear least squares fit. To avoid overfitting effects — like degenerative FFD grids which imply excessive local deformations — the inverse problem is regularized with the Tikhonov approach. The NASA CRM model and the IAE V2500 engine have been selected as reference geometries. Both resemble nacelles that are typically found on common aircraft models and both deviate sufficiently from the rotational symmetry. It is demonstrated that the mean error of the approximation decreases with an increase of the number of FFD grid points and how the regularization affects these results. Finally, the B-spline based FFD with the classical Bernstein based FFD are compared for both models. The results conceptually prove the usability of the FFD approach for the construction of nacelle geometries in the preliminary aircraft design phase.

scholarly journals Finite element based hybrid techniques for advection-diffusion-reaction processes

2018 ◽  
Vol 8 (2) ◽  
pp. 127-136
Author(s):  
Murat Sari ◽  
Huseyin Tunc
Keyword(s):  
Exact Solution ◽  
Finite Element ◽  
Numerical Solutions ◽  
Diffusion Reaction ◽  
Optimization Approach ◽  
B Spline ◽  
Hybrid Techniques ◽  
Advection Diffusion ◽  
Pure Diffusion ◽  
Spline Basis

In this paper, numerical solutions of the advection-diffusion-reaction (ADR) equation are investigated using the Galerkin, collocation and Taylor-Galerkin cubic B-spline finite element method in strong form of spatial elements using an ?-family optimization approach for time variation. The main objective of this article is to capture effective results of the finite element techniques with B-spline basis functions under the consideration of the ADR processes. All produced results are compared with the exact solution and the literature for various versions of problems including pure advection, pure diffusion, advection-diffusion, and advection-diffusion-reaction equations. It is proved that the present methods have good agreement with the exact solution and the literature.

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