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How to design an optimal topology of an inverter mechanism without any hinge? The top picture shows the design domain and boundary conditions. The desired output displacement is opposed to the direction of the applied force. This paper proposed an optimization algorithm to find the optimal topology of the invert mechanism as shown in the below picture. The deformed shape shows that the desired output displacement is successfully obtained through elastic deformation of the hinge-free compliant mechanism.

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... In our own investigations it was observed that this -in combination with a suitable volume constraint -can reduce the occurrence of compliant regions of small dimensions. This observation coincides with those of [34,35]. The object of investigation of this paper is to determine whether the volume constraint is a suitable parameter for designing CMs with distributed compliance. ...

... Here, represents the element strain energies sorted by size and not satisfying condition (35). To ensure that the value obtained is between 0 and 1 regardless of the number of elements, the Gini index is then normalized [43]: ...

... Here, the maximum value = 1 is shown in black. The element strain energies calculated and satisfying condition (35) were normalized for the plot so that the maximum value equals one. To make the distribution more visible, the maximum value of the scale (red) was set to the value 0.7. ...

Many approaches have been developed for the synthesis of compliant mechanisms via topology optimization. Most of these approaches produce mechanisms with lumped compliance if not extended appropriately, such as with stress constraints. With mechanisms with lumped compliance, the deformation is concentrated in regions of small dimensions (referred to as “de facto hinges” or “one-node connected hinges”) causing notch overstresses. Some extensions to these approaches overcome this by forcing mechanisms with distributed compliance (also referred to as “hinge-free compliant mechanisms”). However, the currently known extensions have a number of drawbacks. Stress constraints, for example, make the optimization formulation highly nonlinear and non-convex. In this paper, a new extension is presented that permits the synthesis of mechanisms with distributed compliance by combining a constraint for the allowed structural stiffness with an adaptive volume constraint. The presented optimization approach can be easily solved using linear optimization. The basis for the extension presented in this paper is the modal synthesis approach, which permits the synthesis of transverse-load-insensitive compliant mechanisms. The methodology developed is tested on suitable design examples.

... The Evolutionary topology optimization or ESO method is based on a simple concept that inefficient material is gradually removed from the design domain so that the resulting topology evolves towards an optimum. The later version of the BESO method, namely the bi-directional evolutionary topology optimization (BESO), allows not only to remove material from the least efficient regions but also to add material near the most efficient regions simultaneously (Li et al., 2013). The BESO method could be used in several applications, such as problems with fluid structure interaction (Vicente et al., 2015), design dependent loads (Picelli et al., 2014) or porous-acoustic absorbing systems (Silva and Pavanello, 2010). ...

... The traditional BESO method is used in this work for the compliant mechanism design using a multi-criteria objective function. For this purpose, the structure strain energy is introduced into the optimization problem formulation as was already made by Li et al. (2013). His work is one of the few investigations that used the BESO method for compliant mechanism design and was a very important reference for the development of this investigation. ...

... The BESO algorithm for compliant mechanism design was implemented using MATLAB and validated by comparing the results with topologies found from previous works. For this verification stage, the results from Li (2014) and Li et al. (2013) research are used as a reference, given that, as was already mentioned this was one of the few works to apply the BESO method for compliant mechanisms design. ...

Compliant mechanisms are mechanical devices that achieve motion through elastic deformation. They represent a relatively new variety of lightweight structures and their use has been expanded given their advantages over the traditional rigid-body mechanisms. This paper explores a strategy for compliant mechanisms design using the bi-directional evolutionary topology optimization (BESO) method. This procedure requires choosing an objective function that handles motion and loading requirements simultaneously for a given set of input force and desired output displacements. The objective function maximizes the desirable displacement and reduces the formation of hinges at the same time. Sensitivity numbers are found by the objective function variation with respect to the design variables, to be later used in the BESO method to add and remove material until an optimal topology is achieved. The implementation is validated by comparing the results with typical topologies found from previous works. Finally, the influence of the workpiece constraint out over the final topologies is analyzed. The method could be extended to fluid-actuated mechanisms design by considering pressure loads instead of point loads. This approach will require the implementation of design dependent loads to formulate the optimization problem, an interesting subject to address for further investigations. Keywords: Topology Optimization, BESO, Compliant mechanisms, Multi-criteria Optimization.

... The design of compliant mechanisms using topology optimization usually encounters hinge zones, which can cause high stress concentrations and fabrication difficulties. One can obtain hinge-free compliant mechanisms by formulating this type of topology optimization problem as [43] ⎧ ...

... and the derivatives of the output displacement out and the input displacement in can be found in Li et al. [43]. Based on the traditional SIMP model, the derivative of the total strain energy SE with respect to the design variable ρ i can be easily derived by the adjoint method as ...

This paper proposes a new method for topology optimization of two-dimensional (2D) continuum structures by combining the features of the guide-weight criterion and the conventional bidirectional evolutionary structural optimization (BESO) method. The distribution of material is dominated by guide weights instead of sensitivity numbers. Benefitting from high computational efficiency and the existence of intermediate design variables of the guide-weight criterion, this new algorithm further improves the convergence speed and stability of the objective function. Several typical topology optimization examples of 2D continuum structures are used to demonstrate the efficiency of the proposed method. Numerical results show that convergent, mesh-independent and nearly black-and-white solutions can be achieved and that the proposed method is more stable and efficient than the conventional BESO method.

... Great effort s have been made by various researchers who attempted to remove the de facto hinges to obtain CMs with distributed compliance. These effort s can be broadly classified into the following five categories: (I) second-stage design operation strategy [164][165][166] , (II) modified finite element discretization [84,109,[167][168][169][170] , (III) modified optimization procedure [42,[171][172][173][174][175][176] , (IV) imposition of length scale control [99][100][101][177][178][179][180][181][182][183][184][185] , and (V) hinge-free objective functions [159][160][161]171,[186][187][188][189][190][191][192][193][194] . 9. Topology-optimized compliant displacement inverters that are free of de facto hinges. ...

... Li et al. [191] ; Huang et al. [196] min ...

Compliant mechanisms have become an important branch of modern mechanisms. Unlike conventional rigid body mechanisms, compliant mechanisms transform the displacement and force at least partly through the deformation of their structural components, which can offer a great reduction in friction, lubrication and assemblage. Therefore, compliant mechanisms are particularly suitable for applications in microscale/nanoscale manipulation systems. The significant demand of practical applications has also promoted the development of systematic design methods for compliant mechanisms. Several methods have been developed to design compliant mechanisms. In this paper, we focus on the continuum topology optimization methods and present a survey of the state-of-the-art design advances in this research area over the past 20 years. The presented overview can be helpful to those engaged in the topology optimization of compliant mechanisms who desire to be apprised of the field’s recent state and research tendency.

... Topology optimization [12] is regarded as a mathematical programming method, which offers a scientific and effective manner for finding the optimal material layout in the design domain and accepts a wide range of applications [13][14][15][16][17][18][19]. The homogenization theory is proposed to evaluate effective macroscopic property of architected materials using the information from the micro-architecture [20]. ...

... min is the minimal value of the nodal densities. u is the field of the displacement response in the micro-architecture, which have to satisfy the periodic boundary formulation defined in Eq. (13). u is the virtual displacement field, which belongs to the admissible displacement space H 1 (Ω) with the Y-periodicity. ...

Architected materials with the desirable properties, such as the lightweight and the superior stiffness, have accepted considerable attention in recent years. This paper aims to study the design of architected materials but with the ultra-lightweight using an effective and efficient Isogeometric Topology Optimization (ITO) method. An enough smooth and continuous Density Distribution Function (DDF) is constructed using the Shepard function and NURBS basis functions, to describe the topology of the micro-architecture. Later, the homogenization with a simple periodic boundary formulation is numerically implemented by isogeometric analysis (IGA) to predict effective macroscopic properties using the micro information. An ITO formulation with the specific objective function is developed to rationally design 2D and 3D architected materials with the extreme elastic properties using the DDF and IGA-based homogenization, in which the volume fraction needs to be very low to ensure the optimized micro-architectures with the ultra-lightweight. NURBS basis functions offer a unified formula for the structural geometry, the solution space and the micro topology in architected materials. Finally, several numerical examples are provided to display the effectiveness of the ITO method for the ultra-lightweight architected materials, particularly for 3D scenario. A series of novel and interesting 3D ultra-lightweight architected materials are found, which are also prototyped using the Selective Laser Sintering (SLS) technique.

... Most of these methods are based on topological optimisation specifically on the structure, beam shape, and beam thickness [13][14][15][16][17][18][19]. Compliant mechanisms have superior features such as repeatability and non-existence of backlash, compact, easy manufacturing and low cost [20][21][22][23]. ...

Safe tool-tissue interaction is critical in medicine. The exerted force or torque on the tissue is required to be under control. This paper presents a design framework for producing simple mechanisms with adjustable constant force or torque. The constant load is generated by paralleling a positive constant stiffness spring and a negative constant stiffness mechanism. The combined output load can be tuned by simply changing the pre-load of the positive stiffness spring. Also, an algorithm is proposed as a key component of the design framework for assisting the design of the negative stiffness mechanism. By determining a prescribed stiffness that meets the requirement of the application, the proposed algorithm which consists of a Finite Element Simulation (FES) and Genetic Algorithm (GA) seeks a proper beam structure through iterative optimisation automatically. Two example applications are provided to demonstrate the effectiveness of using the design method in satisfying medical needs. Specifically, one translational application and one rotational application are used to show the capability and the versatility of the design framework. According to the experimental results of both examples, the produced mechanisms are able to output a required constant load along the target displacement consistently, and the output load magnitude can be controlled online.

... These hinge regions are difficult to fabricate especially for the microscale systems and further lead to stress concentration. As summarized by Zhu et al. (2020), this de facto hinge problem can be addressed by (I) second-stage design (Shih and Lin, 2006), (II) modified finite element discretization (Kim et al. 2005), (III) considering stress constraint (De Leon et al. 2015), (IV) hinge-free objective function (Rahmatalla and Swan, 2005;Zhu et al. 2014;Li et al. 2014;Niu et al. 2020), and (V) minimum length scale control. Among these methods, controlling the minimum length scale has a significant place, because it can yield mesh-independent and checkerboard-free topologies and can further improve manufacturability (Lazarov et al. 2016). ...

The discrete variable topology optimization method based on Sequential Approximate Integer Programming (SAIP) and Canonical relaxation algorithm demonstrates its potential to solve large-scale topology optimization problem with 0–1 optimum designs. However, currently, this discrete variable method mainly applies to the minimum compliance problem. The compliant mechanism design is another widely studied topic with distinguishing features. First, the objective function for the compliant mechanism design is non-monotonic with the material usage. Second, since de facto hinges always occur, the minimum length scale control is indispensable for manufacturability. These two issues are well studied in the SIMP approach but bring great challenges when topology optimization problems are formulated in the frame of discrete variables. The present paper generalizes this discrete variable method for the compliant mechanism design problems with minimum length scale control. Firstly, the sequential approximate integer programming with trust region (SAIP-TR) framework is proposed to directly restrict the variation of discrete design variables. Different from the continuous variable optimization, the non-linear trust region constraint can be formulated as a linear constraint under the SAIP framework. By using a merit function, two different trust region adjustment strategies that can self-adaptively adjust the precision of the sub-problems from SAIP-TR are explored. Secondly, a geometric constraint to control the minimum length scale for the material phase and void phase in the framework of discrete design variables is proposed, which suppresses de facto hinges and reduces stress concentration in optimum design. The related issue of feasibility of sub-problem is discussed. By combining the SAIP-TR framework with the geometric constraint, some different hinge-free compliant mechanism designs are successfully obtained.

... Jin and Zhang [27] presented a method for the topology optimization of planar multiple degrees of freedom compliant mechanisms. Li et al. [28] developed a bi-directional structural optimization method for the design of hinge-free compliant mechanisms. Swartz and James [29] proposed Gaussian layer connectivity parameterization technique to optimize multi-body mechanisms. ...

In micro/nano manipulation mechanisms, the compliant z/tip/tilt stages have proved to have enormous advantages for out-of-plane positioning. Small workspace is a challenge in designing these out-of-plane positioning systems. This deficiency can be overcome significantly by a new insight into the optimization approach of a compliant mechanism that can be implemented easily on the entire structure of any spatial mechanism and/or any spatial mechanism with n-symmetrical spatial kinematic chains. Finite Element Method (FEM) adopted by the engineering software, ANSYS, is utilized to perform the structural optimization of an entire three-way symmetric spatial compliant mechanism to achieve the best performances and overcoming deficiencies. Therefore, this paper introduces a compliant monolithic three degrees of freedom (DOFs) z/tip/tilt piezo-driven micromanipulator with a comparable large workspace. Furthermore, as compact as 201 mm × 180 mm × 75 mm and made of Acrylonitrile Butadiene Styrene (ABS) with the first natural frequency of 115.42 Hz make this design superior in comparison to its peers. Regarding the structural aspects, three Scott-Russell and six leaf parallelogram mechanisms are employed to transform horizontal piezoelectric actuators’ inputs (PEAs) to vertical inputs and transfer them to the stage. The best performances of the proposed structure are achieved by optimizing the mechanism using FEM. Due to the optimization, a large amplification ratio of 5.3 is achieved. Moreover, finite element analyses and analytical modeling are performed to verify the performance of the proposed mechanism. Finally, in a comparative study, the captured workspace and resonant frequency of the proposed micromanipulator are compared against those available in theliterature and provide an insight into the performances of the proposed monolithic mechanism.

... Li et al. [109] used a combination of SIMP and ESO. Li et al. [110] used BESO method in the design of hinge-free compliant mechanisms. Maleki Jebeli and Shariat Panahi [111] used GA as evolutionary algorithm to optimize the material property distribution in FG structures. ...

... Numerous TO methods, including homogenization methods [3], solid isotropic material with penalization [4][5], evolutionary structural optimization (ESO) [6][7][8][9], level-set methods [10][11][12][13][14][15], have evolved significantly over the past few decades. Because of its effectiveness and simplicity, ESO-type methods have been successfully applied to solve various optimization problems, such as heat conduction structure design [16], manufacturable lattice structure design [17], reinforced concrete deep beam design [18], hinge-free compliant mechanism design [19], material microstructure design [20], and many other advanced structure design problems. ...

Uncertainty factors play an important role in the design of periodic structures because structures with small periodic design spaces are extremely sensitive to loading uncertainty. Therefore, for the first time, this paper proposes a framework for robust topology optimization (RTO) of periodic structures assuming that load uncertainties follow a Gaussian distribution. In this framework, the expected value and variance of structural compliance can be easily computed using a semi‐analytical method combined with probability theory, which is important for RTO when uncertain variables follow probabilistic distributions. To obtain optimal topologies, the bi‐directional evolutionary structural optimization method is used. Structural periodicity is calculated using a strategy of sensitivity averaging and consistency constraints. To eliminate the influence of numerical units when comparing the optimal results to deterministic and RTO solutions, a generic coefficient of variation is defined as the robust index, which contains both the expected value and variance. The proposed framework is verified through the optimization of both 2D and 3D structures with periodicity. Computational results demonstrate the feasibility and effectiveness of the proposed framework for designing robust periodic structures under loading uncertainties.

... A widely used synthesis approach to CMs is topology optimization (TO) [8] . Many research efforts to optimization schemes [8,9] , analysis approaches [10] , discretization schemes [11][12][13][14] , and numerical difficulties [15][16][17] in topology optimization have been devoted. In TO, CMs are traditionally designed based on a stiffness-flexibility criterion [8] . ...

This paper presents a new topology optimization technique for the design of compliant mechanisms that are efficient in transferring motion, force, or energy while being sufficiently strong to resist yield or fatigue failure. Generally, flexure hinges are efficient in transferring motion, force, or energy but are weak in resisting yield or fatigue failure while slender beams are relatively inefficient but strong. Thus, our philosophy is that a compliant mechanism may benefit from the above complimentary properties of flexure hinges and slender beams if they are connected and sized in a rational way. This requires a design approach with both flexure hinges and beams as constructional elements, and the design approach should include criteria on both efficiency and strength. Therefore, in the proposed technique, a mixed mesh of flexure hinges and beams was employed to discretize the design domain, and their connectivity, locations, and sizes were simultaneously determined to fulfill both the so-called stiffness-flexibility criterion (for efficiency) and a newly proposed input stroke criterion (for strength). The input stroke of a compliant mechanism, defined per the von-Mises yield criterion, is inversely proportional to the maximum stress per input displacement and represents the mechanism's maximum input displacement before yield failure. Both theoretical explorations and design examples demonstrate that the strength of compliant mechanisms can be significantly improved without compromising the efficiency, and trade-off designs that are better balanced between strength and efficiency can be obtained.

... The benefit of topology optimization is that the initial selection of a proper mechanism topology is not required. Many recent research efforts to the optimization formulations [15][16][17][18], discretization schemes [19][20][21][22][23], and numerical difficulties [9,[24][25][26] in topology optimization have been devoted. ...

This paper proposes a topology optimization framework to design compliant mechanisms with a mixed mesh of both beams and flexure hinges for the design domain. Further, a new type of finite element, i.e., super flexure hinge element, was developed to model flexure hinges. Then, an investigation into the effects of the location and size of a flexure hinge in a compliant lever explains why the point-flexure problem often occurs in the resulting design via topology optimization. Two design examples were presented to verify the proposed technique. The effects of link widths and hinge radii were also investigated. The results demonstrated that the proposed meshing scheme and topology optimization technique facilitate the rational decision on the locations and sizes of beams and flexure hinges in compliant mechanisms.

... The basic element of the optimization process is a good way to choose the design parameterization [7] [8]. ...

This paper concerns a structural optimization problem for a flextensional piezoelectric actuator which consists of the high power piezoelectric stack and polymeric composite shell intended for amplification of the stroke. The principal drawback of the piezoelectric transducers is a very small stroke at relatively high operating force. In order to supply the required stroke some amplification means are used. The considered flextensional transducer allows obtaining needed stroke amplification, but because of counteracting forces the initial actuator’s stroke significantly reduced. The main aim of this article is the optimization of the actuator design to simultaneously provide sufficient stroke and stiffness, allowing counteract external loads. To do this, we parameterize the shape of the amplificator’s shell by the rational Bezier curves, which parameters (coordinates and weights of the control points) are changed iteratively by genetic algorithm according to the fitness function value calculated by the finite element model of the transducer with varied geometry of shell.

In this paper, a topology optimization method of compliant mechanisms considering geometrical nonlinearity is proposed. In the method, the HMPS method, which is one of particle methods, is used for the analysis considering geometrical nonlinearity. The CA-IESO (Cellular Automaton - Improved Evolutionary Structural Optimization) method is used for the topology optimization. In the formulations, we show the objective function and the sensitivity those are adapted to the topology optimization of compliant mechanism using the CA-IESO method. Several numerical examples are provided to demonstrate the effectiveness of the proposed method for the topology optimization of compliant mechanisms.

Compared with the strongly-coupled parallel mechanism, the weakly-coupled mechanism can realize two kinds of mutually independent motions. In this paper, a new method for type synthesis of weakly-coupled compliant parallel mechanisms is proposed. The synthesis conditions, that is, the synthesis conditions of the shared flexure hinges and synthesis condition of the weakly-coupled, are given. The synthesis process is formulated. From the perspective of mechanism innovation, a new synthesis idea of weakly-coupled CPMs is introduced. New weakly-coupled CPMs without intermediate platform are obtained by sharing the flexure hinges of two compliant parallel mechanisms. To obtain five degree of freedom (5-DOFs) weakly-coupled compliant parallel mechanisms, two kinds of sub-sets of compliant mechanism blocks, a two-rotation and one-translation compliant mechanism and a two-translation compliant mechanism are proposed. With the proposed synthesis method, two blocks are indirectly connected by sharing flexure hinges to obtain an integral mechanism. Then, new 5-DOF weakly-coupled compliant parallel mechanisms are proposed by the synthesis conditions. Additionally, finite element simulation is carried out to verify the correctness of the proposed synthesis method. Thirteen new 5-DOF weakly-coupled compliant parallel mechansims have been synthesized, which enriches the compliant parallel mechanism configurations. And, they have simple structure, compact structure, ease of control and light weight. The synthesized 5-DOF weakly-coupled compliant parallel mechanisms have potential application to pointing, vibration isolation and mirror adjustment platform.

Compliant mechanisms provide guided motion by utilizing the elastic deformation of their constitutive beams. These beams are usually stiffened via an intermediate element to help adjusting the stiffness in the motion as well as the bearing directions. For the sake of simplicity, the rigidity of this intermediate element has always been assumed to be infinite, while in practical situations, it has a finite compliance. So the objective of this paper is to study the nonlinear static load-displacement relationships of beam-based flexure modules with an intermediate semi-rigid element. The principle of virtual work along with nonlinear von-Karman expression of the strain energy is utilized to obtain the differential equations governing the nonlinear static behavior of a flexure beam with intermediate stiffener as well as the corresponded kinematic and natural boundary conditions. These equations are then solved analytically and closed form expressions are provided for the end point displacement of the beam in terms of the end forces and moments applied to the beam. Also a closed form expression is derived for the nonlinear strain energy of the beam in terms of its end point displacements. These expressions are utilized to model the nonlinear load-displacement behavior of a general parallelogram flexure module. The presented models are verified using nonlinear finite element simulations and excellent agreements are observed. The resulting simulations reveal that lack of consideration of the flexibility of the intermediate elements in flexure modules may induce significant inaccuracies in prediction of the error motions and stiffness of compliant mechanisms. The resulting knowledge from this effort is expected to provide a more precise frame work for the static and dynamic analysis of more complex flexure modules and provide the designers with more optimum design options for the compliant mechanisms.

We introduce a globally convergent sequential linear programming method for nonlinear programming. The algorithm is applied to the solution of classic topology optimization problems, as well as to the design of compliant mechanisms. The numerical results suggest that the new algorithm is faster than the globally convergent version of the method of moving asymptotes, a popular method for mechanical engineering applications proposed by Svanberg.

IntroductionProblem Statement and Material Interpolation SchemeSensitivity Analysis and Sensitivity NumberExamplesConclusion
Appendix 4.1References

This work presents an element addition strategy for 3D compliant mechanisms design. The proposed procedure is based on an extension of the evolutionary structural optimization (ESO) method, which has been successfully applied to several optimum material distribution problems, but not for 3D compliant mechanisms optimization.Even if most investigations for compliant mechanism design have been oriented for planar systems design, this technology may be useful also for 3D mechanisms design, for instance in making devices for micro- and nanomanipulation, like the popular hexapods mechanisms used for six axis positioning. These 3D structures and mechanisms (rigid or compliant) must be carefully manufactured and assembled from many precision components, and there are still many aspects that must be examined to accomplish the topology optimization and ensure the performance of these precision manipulators. The present paper aims to progress on this line, and will apply an alternative approach derived in this investigation, which improves the solutions obtained by this specific method. The proposed method has been tested in several numerical applications and benchmark examples to illustrate and validate the approach, and satisfactorily applied to the solution of 3D examples.

This paper presents an improved algorithm for the bi-directional evolutionary structural optimization (BESO) method for topology optimization problems. The elemental sensitivity numbers are calculated from finite element analysis and then converted to the nodal sensitivity numbers in the design domain. A mesh-independency filter using nodal variables is introduced to determine the addition of elements and eliminate unnecessary structural details below a certain length scale in the design. To further enhance the convergence of the optimization process, the accuracy of elemental sensitivity numbers is improved by its historical information. The new approach is demonstrated by solving several compliance minimization problems and compared with the solid isotropic material with penalization (SIMP) method. Results show the effectiveness of the new BESO method in obtaining convergent and mesh-independent solutions.

The article presents a methodology for generating topology of complaint mechanisms using nonlinear deformation theory. In an earlier work, (Joo, J., Kota, S., Kikuchi, N., (200013.
Joo , J. ,
Kota , S. and
Kikuchi , N. 2000. Topological synthesis of compliant mechanisms using linear beam elements. Mechanics of Structures and Machines, 28(4): 245–280. [CROSSREF] [Taylor & Francis Online], [Web of Science ®], [CSA]View all references) Topological synthesis of compliant mechanisms using linear beam elements. Mechanics of Structures and Machines 28(4):245–280), we presented a topology synthesis methodology using linear beam elements. Using large deformation analysis, the article presents a methodology for topology synthesis based on a multicriteria objective function as a ratio of geometrical advantage to strain energy is employed to capture the conflicting functional requirements that are inherent in compliant mechanisms design. The benefits of using nonlinear methods for large deformation problems are illustrated by using three design examples and comparing results from a nonlinear implementation of the optimization procedure with a linear scheme.

This paper presents a method for optimal design of compliant mechanism topologies. The method is based on continuum-type topology optimization techniques and finds the optimal compliant mechanism topology within a given design domain and a given position and direction of input and output forces. By constraining the allowed displacement at the input port, it is possible to control the maximum stress level in the compliant mechanism. The ability of the design method to find a mechanism with complex output behavior is demonstrated by several examples. Some of the optimal mechanism topologies have been manufactured, both in macroscale (hand-size) made in Nylon, and in microscale (<.5mm)) made of micromachined glass.

Layout or topology optimization deals with the selection of the best configuration for structural systems and constitutes one of the newest and most rapidly expanding fields of structural design, although some of its basic concepts were established almost a century ago. While mathematically and computationally perhaps the most challenging, it is also economically the most rewarding design task. This review article is based on a unified formulation and covers in detail both exact, analytical methods and approximate, discretized methods of layout optimization. Although discretized solutions are unavoidable for most practical, real-world problems, only explicit analytical solutions provide (i) a reliable means for checking the validity and convergence of numerical methods and (ii) a basis for assessing the relative economy of other designs. Moreover, some of the most efficient new numerical methods of layout optimization are iterative versions of analytical methods. Particularly promising are recent extensions of the exact layout theory to multiload, multipurpose elastic systems.

There are several well-established techniques for the generation of solid-void optimal topologies such as solid isotropic
material with penalization (SIMP) method and evolutionary structural optimization (ESO) and its later version bi-directional
ESO (BESO) methods. Utilizing the material interpolation scheme, a new BESO method with a penalization parameter is developed
in this paper. A number of examples are presented to demonstrate the capabilities of the proposed method for achieving convergent
optimal solutions for structures with one or multiple materials. The results show that the optimal designs from the present
BESO method are independent on the degree of penalization. The resulted optimal topologies and values of the objective function
compare well with those of SIMP method.

While much has been contributed to techniques for enumerating and identifying rigid-body mechanisms in the past decades, proportionally little has been accomplished in this regard in compliant mechanisms design. This paper deals primarily with indentification and discussion of important kinematic properties of compliant mechanisms. To facilitate these appropriate terminology is developed at the very fundamental level. The conventional degrees-of-freedom concept for a rigid-body chain is briefly reviewed. It is then used to help define a compliance number (or degrees-of-compliance) concept for characterizing compliant mechanisms. Finally, a systematic and convenient approach is presented, enabling the type synthesis of this class of mechanisms.

A simple evolutionary procedure is proposed for shape and layout optimization of structures. During the evolution process low stressed material is progressively eliminated from the structure. Various examples are presented to illustrate the optimum structural shapes and layouts achieved by such a procedure.

Compliant or flexible-link mechanisms gain some or all of their motion from the relative flexibility of their joints rather than from rigid-body joints only. Unlike rigid-body mechanisms, energy is not conserved between the input and output ports of compliant mechanisms because of energy storage in the flexible members. This effect and the nonlinearities introduced by large deflections complicate the analysis of such mechanisms. The design of compliant mechanisms in industry is currently accomplished by expensive trial and error methods. This paper introduces a method to aid in the design of a class of compliant mechanisms wherein the flexible sections (flexural pivots) are small in length compared to the relatively rigid sections. The method includes a definition and use of a pseudo-rigid-body model, and the use of a large-deflection finite element type algorithm. An example is used to illustrate the design technique described.

Compliant mechanisms gain at least some of their motion from flexible members. The combination of large-deflection beam analysis, kinematic motion analysis, and energy storage makes the analysis of compliant mechanisms difficult. The design of mechanisms often requires iteration between synthesis and analysis procedures. In general, the difficulty in analysis has limited the use of compliant mechanisms to applications where only simple functions and motions are required. The pseudo-rigid-body model concept promises to be the key to unifying the compliant and rigid-body mechanism theories. It simplifies compliant mechanism analysis by determining an equivalent rigid-body mechanism that accurately models the kinematic characteristics of a compliant mechanism. Once this model is obtained, many well known concepts from rigid-body mechanism theory become amenable for use to analyze and design compliant mechanisms. The pseudo-rigid-body-model concept is used to develop a loop-closure method for the analysis and synthesis of compliant mechanisms. The method allows compliant mechanisms to be designed for tasks that would have earlier been assumed to be unlikely, if not impossible, applications of compliant mechanisms.

The design of compliant mechanisms poses certain unique challenges because such mechanisms should have adequate flexibility to undergo desired deformations under the action of applied forces and adequate stiffness to withstand external loading. The focus here is to generate the topology of a compliant mechanism starting from input/output force/displacement functional requirements and design constraints. Previous studies [[1]1.
Ananthasuresh , G. K. , Kota , S. and Gianchandani , Y. 1993 . Systematic Synthesis of Microcompliant Mechanisms—Preliminary Results . Proc. 3d Natl. Conf. on Applied Mechanisms and Robotics . November 1993 , Cincinnati. Vol. 2 , View all references [2]2.
Ananthasuresh , G. K. , Kota , S. and Gianchandani , Y. June 1994 . “ A Methodical Approach to the Synthesis of Micro Compliant Mechanisms ” . In Technical Digest, Solid-State Sensor and Actuator Workshop 189 – 192 . Island, S. C. : Hilton Head . View all references [3]3.
Ananthasuresh , G. K. , Kota , S. and Kikuchi , N. Strategies for Systematic Synthesis of Compliant MEMS, DSC . ASME Winter Annual Meeting . Nov. 1994 , Chicago. Vol. 55-2 , View all references] and [[4]4.
Frecker , M. I. , Ananthasuresh , G. K. , Nishiwaki , S. , Kikuchi , N. and Kota , S. 1997 . Topological Synthesis of Compliant Mechanisms Using Multi-Criteria Optimization . J. Mech. Design , 119 ( 2 ) : 238 – 245 . [CrossRef], [Web of Science ®]View all references] employed a multi-criteria objective function comprised of mutual potential energy (MPE) and strain energy (SE) to full ground truss structures. Here an improved and robust objective function and its implementation for a network of linear beam elements is presented. Also discussed is the influence of various geometric and material variables on the objective function. Additionally, the objective function is interpreted in terms of physical design parameters such as mechanical advantage and geometric advantage.

A formulation for design of continuous, hinge-free compliant mechanisms is developed and examined within a continuum structural topology optimization framework. The formulation makes use of two distinctly different sets of springs, the first of which are artificial springs of relatively large stiffness attached to the input and output ports of the mechanism model, and the second of which are springs attached only to the output port with smaller stiffnesses that represent the resistance of the workpiece as it is manipulated by the mechanism. The proposed formulation involves solving two nested optimization problems. In the inner problem the arrangement of a constrained amount of structural material is optimized to maximize the mechanism's mutual potential energy in response to a force loading at the input port while working against the stiff artificial springs on the input and output ports. As the relative stiffness of the artificial springs increases, the material continuity of the mechanism also increases to the point where de facto ‘hinge’ regions are eliminated. In the outer problem, the artificial springs are removed and one solves for an appropriate amount of structural material that yields the desired finite deformation compliance characteristics of the mechanism when working against the real workpiece resistance. Different aspects of the proposed formulation are demonstrated on a number of examples and discussed. Copyright © 2004 John Wiley & Sons, Ltd.

The paper deals with topology optimization of structures undergoing large deformations. The geometrically nonlinear behaviour
of the structures are modelled using a total Lagrangian finite element formulation and the equilibrium is found using a Newton-Raphson
iterative scheme. The sensitivities of the objective functions are found with the adjoint method and the optimization problem
is solved using the Method of Moving Asymptotes. A filtering scheme is used to obtain checkerboard-free and mesh-independent
designs and a continuation approach improves convergence to efficient designs.
Different objective functions are tested. Minimizing compliance for a fixed load results in degenerated topologies which are
very inefficient for smaller or larger loads. The problem of obtaining degenerated “optimal” topologies which only can support
the design load is even more pronounced than for structures with linear response. The problem is circumvented by optimizing
the structures for multiple loading conditions or by minimizing the complementary elastic work. Examples show that differences
in stiffnesses of structures optimized using linear and nonlinear modelling are generally small but they can be large in certain
cases involving buckling or snap-through effects.

Anion-exchange superporous cellulose (DEAE-SC) and microporous cellulose (DEAE-MC) adsorbents were packed in an electrochromatographic
column, and the effect of external electric field (eEF) on the dynamic adsorption was investigated. The column was designed
to provide longitudinal, transverse or 2-dimensional (2D) eEF. It was found that the electro-kinetic effect caused by the
introduction of an electric field played an important role in the dynamic adsorption of bovine serum albumin to the adsorbents.
The dynamic binding capacity (DBC) in the presence of 2D eEF was higher than in the presence of a one-dimensional eEF. The
effect of flow velocity on the DBC of the two adsorbents was also demonstrated. It was found that the effect of electric field
on the DEAE-MC column was more remarkable than that on the DEAE-SC column at the same flow rate, whereas the DEAE-SC column
showed higher DBC and adsorption efficiency (AE) than the DEAE-MC column. With increasing flow rate, the DEAE-SC column could
still offer high DBC and AE in the presence of the 2D eEF. For example, a DBC of 21.4 mg/mL and an AE of 57.7% were obtained
even at a flow rate as high as 900 cm/h. The results indicate that the 2D electrochromatography packed with the superporous
cellulose adsorbent is promising for high-speed protein chromatography.

An optimal structural property for compliant topologies is presented in this paper for general multicriteria formulations
that comprise the conflicting flexibility and stiffness requirements. The property deduced from the first-order necessary conditions for optimality implies that the ratio of the mutual potential energy density to the strain energy density is uniform throughout the continuum, but for
portions otherwise bounded by gage constraints. This property is used to develop an optimality criteria method for synthesizing compliant topologies. It is also noted that
the multicriteria formulations considered here are nonconvex and can result in nonunique solutions. However, by incorporating
a one-variable search along the direction determined by the above optimal property, it is ensured that the converged solution
is a minimum. Several synthesis examples are included with linear frame finite elements which are easy for implementation
and are capable of appropriately accounting for the bending behaviour in the continuum. Examples with previously reported
density based design parameterization using bilinear plane-stress elements are also included to illustrate the synthesis procedure.

This paper presents a focused examination of the mechanical and geometric advantages in compliant mechanisms and their ramifications
in the design formulations of compliant mechanisms posed as a topology optimization problem. With a linear elastic structural
analysis, we quantify mechanical (and geometric) advantage in terms of the stiffness elements of the mechanism’s structure.
We then analyze the common formulations of compliant mechanism optimization and the role of the external springs added in
the formulations. It is shown that the common formulations using mechanical (or geometric) advantage would directly emulate
at best a rigid-body linkage to the true optimum design. As a result, the topology optimization generates point flexures in
the resulting optimal mechanisms. A case study is investigated to demonstrate the resulting trends in the current formulations.

This paper presents a simple method for structural optimization with frequency constraints. The structure is modelled by a fine mesh of finite elements. At the end of each eigenvalue analysis, part of the material is removed from the structure so that the frequencies of the resulting structure will be shifted towards a desired direction. A sensitivity number indicating the optimum locations for such material elimination is derived. This sensitivity number can be easily calculated for each element using the information of the eigenvalue solution. The significance of such an evolutionary structural optimization (ESO) method lies in its simplicity in achieving shape and topology optimization for both static and dynamic problems. In this paper, the ESO method is applied to a wide range of frequency optimization problems, which include maximizing or minimizing a chosen frequency of a structure, keeping a chosen frequency constant, maximizing the gap of arbitrarily given two frequencies, as well as considerations of multiple frequency constraints. The proposed ESO method is verified through several examples whose solutions may be obtained by other methods.

This work presents a modified version of the evolutionary structural optimization (ESO) procedure for its application in topology optimization of compliant mechanisms. The proposed procedure is based in the ESO method, which has been successfully applied to several optimum material distribution problems, such us stiffness, frequency or buckling problems, but not for compliant mechanism optimization. It will be shown that an additive version of this method must be adopted in order to achieve the optimum design, since the traditional ESO method's element removal technique is not efficient in this case.The proposed method will apply the objective functions and restrictions suggested in previous works, as well as original formulations and alternative approaches derived in this investigation which improve the solutions obtained by other authors.The procedure has been implemented into a general optimization software and tested in several numerical applications and benchmark examples to illustrate and validate the approach.

After outlining analytical methods for layout optimization and illustrating them with examples, the COC algorithm is applied to the simultaneous optimization of the topology and geometry of trusses with many thousand potential members. The numerical results obtained are shown to be in close agreement (up to twelve significant digits) with analytical results. Finally, the problem of generalized shape optimization (finding the best boundary topology and shape) is discussed.

We devise new numerical algorithms, called PSC algorithms, for following fronts propagating with curvature-dependent speed. The speed may be an arbitrary function of curvature, and the front also can be passively advected by an underlying flow. These algorithms approximate the equations of motion, which resemble Hamilton-Jacobi equations with parabolic right-hand sides, by using techniques from hyperbolic conservation laws. Non-oscillatory schemes of various orders of accuracy are used to solve the equations, providing methods that accurately capture the formation of sharp gradients and cusps in the moving fronts. The algorithms handle topological merging and breaking naturally, work in any number of space dimensions, and do not require that the moving surface be written as a function. The methods can be also used for more general Hamilton-Jacobi-type problems. We demonstrate our algorithms by computing the solution to a variety of surface motion problems.

Optimal shape design of structural elements based on boundary variations results in final designs that are topologically equivalent to the initial choice of design, and general, stable computational schemes for this approach often require some kind of remeshing of the finite element approximation of the analysis problem. This paper presents a methodology for optimal shape design where both these drawbacks can be avoided. The method is related to modern production techniques and consists of computing the optimal distribution in space of an anisotropic material that is constructed by introducing an infimum of periodically distributed small holes in a given homogeneous, isotropic material, with the requirement that the resulting structure can carry the given loads as well as satisfy other design requirements. The computation of effective material properties for the anisotropic material is carried out using the method of homogenization. Computational results are presented and compared with results obtained by boundary variations.

In this paper, a parameterization level set method is presented to simultaneously perform shape and topology optimization of compliant mechanisms. The structural shape boundary is implicitly embedded into a higher-dimensional scalar function as its zero level set, resultantly, establishing the level set model. By applying the compactly supported radial basis function with favorable smoothness and accuracy to interpolate the level set function, the temporal and spatial Hamilton–Jacobi equation from the conventional level set method is then discretized into a series of algebraic equations. Accordingly, the original shape and topology optimization is now fully transformed into a parameterization problem, namely, size optimization with the expansion coefficients of interpolants as a limited number of design variables.Design of compliant mechanisms is mathematically formulated as a general optimization problem with a nonconvex objective function and two additionally specified constraints. The structural shape boundary is then advanced as a process of renewing the level set function by iteratively finding the expansion coefficients of the size optimization with a sequential convex programming method. It is highlighted that the present method can not only inherit the merits of the implicit boundary representation, but also avoid some unfavorable features of the conventional discrete level set method, such as the CFL condition restriction, the re-initialization procedure and the velocity extension algorithm. Finally, an extensively investigated example is presented to demonstrate the benefits and advantages of the present method, especially, its capability of creating new holes inside the design domain.

This paper proposes an innovative, integrated design method for the design of practical and sophisticated compliant mechanisms. The approach consists of two optimisation methods, topology and shape optimisation, plus a scheme to implement designer input of ideas. In the first step, a designer explores the most fruitful design concepts for mechanisms that achieve the design specifications, by combining compliant mechanisms created by the topology optimisation with additional mechanisms prepared by the designer. In this first step, a support method based on the visualisation of the designer's thinking processes assists the designer in his or her exploration of new ideas and design concepts. In the second step, the shape optimisation yields a detailed optimal shape based on the design concept. The combination of compliant mechanisms with the additional mechanisms enables the creation of devices having increased capability or higher performance than would be possible using a single compliant mechanism designed by topology optimisation alone. Executing the shape optimisation after initial design concepts have been explored facilitates the determination of a detailed optimal shape, and also enables to consider non-linear analysis and stress concentration and to make accurate quantitative performance evaluations, which topology optimisation cannot provide.

Compliant mechanisms are single-piece flexible structures that deliver the desired motion by undergoing elastic deformation as opposed to jointed rigid body motions of conventional mechanisms. Compliance in design leads to jointless, no-assembly (Fig. 1), monolithic mechanical devices and is particularly suited for applications with small range of motions. The compliant windshield wiper shown in Fig. 1 illustrates this paradigm of no-assembly. Conventional flexural mechanisms employ flexural joints that connect relatively rigid links as depicted in Fig. 2. Reduced fatigue life, high stress concentration and difficulty in fabrication are some of the drawbacks of flexural joints. Our focus is on designing compliant mechanisms with distributed compliance which employs flexural links (see Fig. 3) and have no joints (neither pin nor flexural joints) for improved reliability, performance, and ease of manufacture. Distributed compliant mechanisms derive their flexibility due to topology and shape of the material continuum rather than concentrated flexion at few regions. This paper focuses on the unique methodology employed to design jointless mechanisms with distributed compliance. The paper also illustrates a compliant stroke amplification mechanism that was recently designed, fabricated and tested for MEMS application. Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/44061/1/10470_2004_Article_353448.pdf

A procedure to obtain a topology of an optimal structure considering flexibility is presented. The methodology is based on a mutual energy concept for formulation of flexibility and the homogenization method. A multi-objective optimization problem is formulated as an application of compliant mechanism design. Some examples of the design of compliant mechanisms for plane structures are presented. © 1998 John Wiley & Sons, Ltd. Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/34532/1/372_ftp.pdf

Seismic Analysis (Chap. 20) Computational Geometry (Chap. 19) Computer Vision (Chaps, 16, 17) Optimality and Control (Chap. 20) Fluid Mechanics (Chap. 18) Combustion (Chap. 18) Materials Sciences (Chap. 18) Semiconductor Manufacturing (Chap. 21) This book is an introduction to level set methods and Fast Marching Methods, which are powerful numerical techniques for analyzing and computing interface motion in a host of settings. They rely on a fundamental shift in how one views moving boundaries, rethinking the natural geometric Lagrangian perspective and replacing it with an Eulerian partial differential equation perspectives. The resulting numerical techniques are used to track three-dimensional fronts that can develop sharp corners and change topology as they evolve. The book begins with an overview of the two techniques, and then provides an int

Compliant mechanisms

- L L Howell

Howell LL. Compliant mechanisms. New York: John Wiley & Sons; 2001.