Continuum Structures Research Articles

Design and analysis of hybrid-driven origami continuum robots with extensible and stiffness-tunable sections

Tendon-driven continuum robots have increasingly attracted attention these years. Conventionally, such kind of robots utilizes elastic central backbones to hold the structure, which makes the robots inextensible, as well as reduces the dexterity and the workspace. Inspired by the reconfigurable feature of origami structures, this paper presents the design, analysis, and validation of a hybrid-driven continuum robot without an elastic backbone. The fabric-based, soft, and unstretchable origami pneumatic chamber holds the continuum structure and makes the robot exhibit a high extension ratio, low input pressure, and no radial expansion. With the antagonistic actuation of tendon-pulling and air-pushing, the robot can perform 3DoF motion with variable stiffness. The kinetostatics modeling and analysis are developed based on a discretization-based approach to predict the motion behavior and control the proposed robot. To validate the proposed design principle and the modeling method, a prototype is built, on which a series of experiments have been conducted. The results show that, with the proposed kinetostatics model, the prototype possesses acceptable positioning accuracy and tracking performances, while the structural stiffness can also be effectively adjusted.

Explicit control of 2D and 3D structural complexity by discrete variable topology optimization method

The structural complexity (the number of holes) of the 2D or 3D continuum structures can be measured by their topology invariants (i.e., Euler and Betti numbers). Controlling the 2D and 3D structural complexity is significant in topology optimization design because of the various consideration, including manufacturability and necessary structural redundancy, but remains a challenging subject. In this paper, we propose a programmable Euler–Poincaré formula to efficiently calculate the Euler and Betti numbers for the 0–1 pixel-based structures. This programmable Euler–Poincaré​ formula only relates to the nodal density and nodal characteristic vector that represents the nodal neighbor relation so that it avoids manually counting the information of the vertices, edges, and planes on the surfaces of the structure. As a result, the explicit formulations between the structural complexity (the number of holes) and the discrete density design variables for 2D and 3D continuum structures can be efficiently constructed. Furthermore, the discrete variable sensitivity of the structural complexity is calculated through the programmable Euler–Poincaré formula so that the structural complexity control problem can be efficiently and mathematically solved by Sequential Approximate Integer Programming and Canonical relaxation algorithm Various 2D and complicated 3D numerical examples are presented to demonstrate the effectiveness of the method. We further believe that this study bridges the gap between structural topology optimization and mathematical topology analysis, which is much expected in the structural optimization community.

Can the Shape of a Planar Pathway Be Estimated Using Proximal Forces of Inserting a Flexible Shaft?

The shape information of flexible endoscopes or other continuum structures, e.g., intro-vascular catheters, is needed for accurate navigation, motion compensation, and haptic feedback in robotic surgical systems. Existing methods rely on optical fiber sensors, electromagnetic sensors, or expensive medical imaging modalities such as X-ray fluoroscopy, magnetic resonance imaging, and ultrasound to obtain the shape information of these flexible medical devices. Here, we propose to estimate the shape/curvature of a continuum structure by measuring the force required to insert a flexible shaft into the internal channel/pathway of the continuum. We found that there is a consistent correlation between the measured insertion force and curvature of the planar continuum pathway. A testbed was built to insert a flexible shaft into a planar continuum pathway with adjustable shapes. The insertion forces, insertion displacement, and the shapes of the pathway were recorded. A neural network model was developed to model this correlation based on the training data collected on the testbed. The trained model, tested on the testing data, can accurately estimate the curvature magnitudes and the accumulated bending angles of the pathway simply based on the measured insertion force at the proximal end of the shaft. The approach may be used to estimate the curvature magnitudes and accumulated bending angles of flexible endoscopic surgical robots or catheters for accurate motion compensation, haptic force feedback, localization, or navigation. The advantage of this approach is that the employed proximal force can be easily obtained outside the pathway or continuum structure without any embedded sensor in the continuum structure. Future work is needed to further investigate the correlation between insertion forces and the pathway and enhance the capability of the model in estimating more complex shapes, e.g., spatial shapes with multiple bends.

A boundary density evolutionary topology optimization of continuum structures with smooth boundaries

AbstractBased on the variable density method, this article proposes a boundary density evolutionary topology optimization method. The method uses a material interpolation model without penalization. Combined with the density grading filtering method, an optimal topology with only 0/1 cells can be obtained. Compared with the solid isotropic microstructures with penalization method (SIMP), no penalty factor is required in the material interpolation model; compared with the evolutionary structural optimization method (ESO), intermediate‐density elements are allowed in the optimization process, but the concept of gradually removing the low‐utilization materials near the boundary in the ESO method is retained. After the optimal result is obtained, the structural boundary element is processed by the level set of nodal strain energy, and the optimization result with smooth boundaries similar to the level set method (LSM) can be obtained. The proposed method has the superiority of the variable density method, and it also combines the advantages of the evolutionary method and the level set method, so which is named as boundary density evolution (BDE) method. The four static and one dynamic optimization examples illustrate the stability and efficiency of the proposed method.

Topology optimization based on the high-order numerical manifold method by implementing a four-node quadrilateral element

In this study, the numerical manifold method (NMM), coupled with material interpolation, is applied to the topology optimization of continuum structures. Quadrilateral element (Q4) meshes are used in the NMM for mathematical cover. The order of the local approximation function is increased to improve the accuracy of the analysis. This method usually leads to linear dependency of the global degrees of freedom, as reflected in the rank deficiency of the global approximation space. However, using Q4 elements, the rank deficiency value is not fixed with mesh refinement. The presented solution handles this problem, and the efficiency of the proposed method is investigated with four examples. The results indicate that the high-order numerical manifold method (HONMM) has great accuracy in the analysis and works well for topology optimization problems. One advantage of using the combined HONMM with material interpolation in topology problems is chequerboard-free patterns, owing to mathematical cover and high-order analysis.

A Gradient-Free Topology Optimization Strategy for Continuum Structures with Design-Dependent Boundary Loads

In this paper, the topology optimization of continuum structures with design-dependent loads is studied with a gradient-free topology optimization method in combination with adaptive body-fitted finite element mesh. The material-field series-expansion (MFSE) model represents the structural topology using a bounded material field with specified spatial correlation and provides a crisp structural boundary description. This feature makes it convenient to identify the loading surface for the application of the design-dependent boundary loads and to generate a body-fitted mesh for structural analysis. Using the dimension reduction technique, the number of design variables is significantly decreased, which enables the use of an efficient Kriging-based algorithm to solve the topology optimization problem. The effectiveness of the proposed method is demonstrated using several numerical examples, among which a design problem with geometry and contact nonlinearity is included.

Finite variation sensitivity analysis for discrete topology optimization of continuum structures

This paper proposes two novel approaches to perform more suitable sensitivity analyses for discrete topology optimization methods. To properly support them, we introduce a more formal description of the Bi-directional Evolutionary Structural Optimization (BESO) method, in which the sensitivity analysis is based on finite variations of the objective function. The proposed approaches are compared to a naive strategy; to the conventional strategy, referred to as First-Order Continuous Interpolation (FOCI) approach; and to a strategy previously developed by other researchers, referred to as High-Order Continuous Interpolation (HOCI) approach. The novel Woodbury approach provides exact sensitivity values and is a better alternative to HOCI. Although HOCI and Woodbury approaches may be computationally prohibitive, they provide useful expressions for a better understanding of the problem. The novel Conjugate Gradient Method (CGM) approach provides sensitivity values with arbitrary precision and is computationally viable for a small number of steps. The CGM approach is a better alternative to FOCI since, for appropriate initial conditions, it is always more accurate than the conventional strategy. The standard compliance minimization problem with volume constraint is considered to illustrate the methodology. Numerical examples are presented together with a broad discussion about BESO-type methods.

Real-Time Dynamics of Cable-Driven Continuum Robots Considering the Cable Constraint and Friction Effect

In this letter, a novel dynamic model of a cable-driven continuum robot is established based on the principle of virtual power, in which the dynamically centralized length of the driving cable is used to simulate the driving process. This loading model implicitly represents the driving loads and explicitly describes the geometrical constraint of cables on the system. The distribution law for friction forces of actuated cables is demonstrated, which can be handily represented by the derived relationship between the tension of the cables and the Lagrange multiplier. This dynamic model is applicable for continuum robots with a single/multi-tandem arm driven by any number of cables. Furthermore, an average stress strategy is developed so that the high-frequency components of the dynamic model can be reasonably controlled in a simple but effective manner, which can strikingly reduce the stiffness of the differential equations. Simulation results show that this strategy significantly improves the simulation efficiency with sound and reasonable precision. A comparison of the results from the numerical simulations and experiments for a cable-driven continuum structure verifies the validity of the proposed model, and the average percentage error for the trajectory between the simulation and experimental results is 1.92%. Moreover, the numerical simulations for the dynamic equations of the cable-driven continuum robot with twenty segments can be run in real-time.

Numerical simulation of stress-based topological optimization of continuum structures under casting constraints

Numerical simulation of stress-based topological optimization of continuum structures under casting constraints

The magneto-electro-elastic multi-physics coupling element free Galerkin method for smart structures in statics and dynamics problems

To solve the problem of over-stiffness and the imprecise magneto-electro-elastic coupling effects of finite element model, we presented the element free Galerkin model to more accurately simulate the statics and dynamics behaviour of magneto-electro-elastic structures. In the element free Galerkin model, the moving least squares approximation was introduced into a magneto-electro-elastic multi-physical-field model. The element free Galerkin model can achieve a close-to-exact stiffness of the continuum structures which could automatically discrete nodes for complicated regions more readily and thus remarkably reduced the numerical errors. In addition, the Runge–Kutta method was presented for solving the motion equation of magneto-electro-elastic structures. Several numerical examples were investigated and exhibited that the element free Galerkin model could receive more accurate and reliable simulation results than the standard finite element model. Besides, the element free Galerkin model was employed to calculate transient responses of magneto-electro-elastic sensor and typical magneto-electro-elastic energy harvester. Therefore, the element free Galerkin model can be adopted to tackle the practical magneto-electro-elastic problems such as smart vibration transducers, magnetic field sensors, and energy harvester devices in intelligent magneto-electro-elastic structures systems.

Robust topology optimization of continuum structures under uncertain partial collapses

This paper presents a novel probabilistic approach for fail-safe robust topology optimization with the following novelties: (1) the probability for failure to occur at a specified location is considered; (2) the possibility for random failure size is incorporated; (3) a multi-objective problem is pursued encompassing both the expected value of the structural performance and its variance as a robustness criterion. Compared against alternative worst-case-based formulations, the probabilistic framework employed allows designers to assume certain level of risk, avoiding undesirable increments in structural performance due to low probability damage configurations; (4) alternatively to most existing works within fail-safe topology optimization, considering density-based methods, this paper pursues for the first time an optimization technique where the structural boundary is represented implicitly by an iso-level of an optimality criterion field, which is gradually evolved using a bisection method. A key advantage of this technique is that it provides optimized solutions for different volume fractions during the optimization process, allowing to efficiently find a trade-off between structural performance, cost and robustness. Finally, numerical results are included demonstrating the ability of the proposed formulation to provide smooth and clearly defined structural boundaries and to enhance structural robustness with respect to conventional deterministic designs.

Nikolai Nikolaevich Luzin at the crossroads of the dramatic events of the European history of the first half of the 20th century

Nikolai Nikolaevich Luzin’s life (1883–1950) and work of this outstanding Russian mathematician, member of the USSR Academy of Sciences and foreign member of the Polish Academy of Arts and Sciences, coincides with a very difficult period in Russian history: two World Wars, the 1917 revolution in Russia, the coming to power of the Bolsheviks, the civil war of 1917–1922, and finally, the construction of a new type of state, the Union of Soviet Socialist Republics. This included collectivization in the agriculture and industrialization of the industry, accompanied by the mass terror that without exception affected all the strata of the Soviet society. Against the background of these dramatic events took place the proces of formation and flourishing of Luzin the scientist, the creator of one of the leading mathematical schools of the 20th century, the Moscow school of function theory, which became one of the cornerstones in the foundation of the Soviet mathematical school. Luzin’s work could be divided into two periods: the first one comprises the problems regarding the metric theory of functions, culminating in his famous dissertation Integral and Trigonometric Series (1915), and the second one that is mainly devoted to the development of problems arising from the theory of analytic sets. The underlying idea of Luzin’s research was the problem of the structure of the arithmetic continuum, which became the super task of his work. The destiny favored the master: the complex turns of history in which he was involved did not prevent, and sometimes even favored the successful development of his research. And even the catastrophe that broke out over him in 1936 – “the case of Academician Luzin” – ended successfully for him.

Three-body description of C9 : Role of low-lying resonances in breakup reactions

The $^9$C nucleus and related capture reaction, ${^8\mathrm{B}}(p,\gamma){^9\mathrm{C}}$, have been intensively studied with an astrophysical interest. Due to the weakly-bound nature of $^9$C, its structure is likely to be described as the three-body (${^7\mathrm{Be}}+p+p$). Its continuum structure is also important to describe reaction processes of $^9$C, with which the reaction rate of the ${^8\mathrm{B}}(p,\gamma){^9\mathrm{C}}$ process have been extracted indirectly. We perform three-body calculations on $^9$C and discuss properties of its ground and low-lying states via breakup reactions. We employ the three-body model of $^9$C using the Gaussian-expansion method combined with the complex-scaling method. This model is implemented in the four-body version of the continuum-discretized coupled-channels method, by which breakup reactions of $^9$C are studied. The intrinsic spin of $^7$Be is disregarded. By tuning a three-body interaction in the Hamiltonian of $^9$C, we obtain the low-lying $2^+$ state with the resonant energy 0.781 MeV and the decay width 0.137 MeV, which is consistent with the available experimental information and a relatively high-lying second $2^+$ wider resonant state. Our calculation predicts also sole $0^+$ and three $1^-$ resonant states. We discuss the role of these resonances in the elastic breakup cross section of $^9$C on $^{208}$Pb at 65 and 160 MeV/A. The low-lying 2$^+$ state is probed as a sharp peak of the breakup cross section, while the 1$^-$ states enhance the cross section around 3 MeV. Our calculations will further support the future and ongoing experimental campaigns for extracting astrophysical information and evaluating the two-proton removal cross-sections.

A novel dynamic reliability-based topology optimization (DRBTO) framework for continuum structures via interval-process collocation and the first-passage theories

In view of the general inertia and damping features as well as the inevitable uncertainty factors in engineering structures, a novel dynamic reliability-based topology optimization (DRBTO) strategy is investigated for time-variant mechanical systems with overall consideration of material dispersion and loading deviation effects. The static interval-set model is first utilized to quantify multi-source uncertainty inputs and the transient interval-process model is then established to characterize unknown-but-bounded response results, which can be readily solved through the proposed interval-process collocation approach combined with a classical Newmark difference scheme. Different from the traditional deterministic design framework, the present DRBTO scheme will directly consider new reliability constraints, for which the non-probabilistic time-variant reliability (NTR) index is mathematically deduced using the first-passage principle. In addition, the issues related to uncertainty-oriented design sensitivity and filtering method are discussed. The usage and effectiveness of DRBTO are demonstrated with three numerical examples.

Optimized pulses for population transfer via laser-induced continuum structures

We use optimal control in order to find the optimal shapes of pulses maximizing the population transfer between two bound states which are coupled via a continuum of states. We find that the optimal bounded controls acquire the bang-interior and interior-bang form, with the bang part corresponding to the maximum allowed control value and the interior part to values between zero and the maximum. Then, we use numerical optimal control to obtain the switching times and the interior control values. We compare our results with those obtained using Gaussian STIRAP pulses, and find that the optimal method performs better, with the extent of improvement depending on the effective two-photon detuning and the size of incoherent losses. When we consider effective two-photon resonance, the improvement is more dramatic for larger incoherent losses, while when we take into account the effective two-photon detuning, the improvement is better for smaller incoherent losses. We also obtain how the transfer efficiency increases with increasing absolute value of the Fano factor. The present work is expected to find application in areas where the population transfer between two bound states through a continuum structure plays an important role, for example coherence effects, like population trapping and electromagnetically induced transparency, optical analogs for light waves propagating in waveguide-based photonic structures, and qubits coupled via a continuum of bosonic or waveguide modes.

Topology optimization of continuum structures under geometric uncertainty using a new extended finite element method

In this article, robust topology optimization under geometric uncertainty is proposed. The design domain is discretized by an extended finite element method. A bi-directional evolutionary structural optimization carries out the optimization process. The performance of the proposed method is compared with the Monte Carlo, solid isotropic material with penalization, perturbation and non-intrusive polynomial chaos expansion methods. The novelty of the present method lies in the following three aspects: (1) this article is among the first to use the extended finite element method in studying the topology optimization under uncertainty; (2) by adopting the extended finite element method for boundary elements in the finite element framework, there is no need for any remeshing techniques; and (3) the numerical results show that the present method has a smoother boundary region and minimum value of the mean and standard deviation of compliance than the other methods, in particular mesh size.

Finite cell method for detection of flaws in plate structures using dynamic responses

Finite cell method (FCM) is an efficient numerical method for analysis of continuum structures with geometrical complexity, which uses the concept of virtual domain method. The use of this concept makes the mesh generation easier such that simple uniform meshes can be employed, and hence mesh refinement is not necessary for the problems like damage detection. Nevertheless, the FCM has not yet been used for damage detection of structures. In this paper, the FCM is proposed for structural damage identification. The identification of flaws based on dynamic response of the structure is investigated with the FCM and particle swarm optimization (PSO) algorithm. The FCM is utilized to model the flawed structures, while the PSO —as a well-known non-gradient-based optimization method— is employed for finding the flaw location. For the damage detection problem, the PSO explores the search space in order to detect the geometrical properties of flaws as well as the number of flaws by minimizing an error function in terms of sensor measurements. Four numerical examples including benchmark test cases are provided to demonstrate the capability of the FCM and the PSO algorithm in damage detection of plate structures.

Dynamic Topology Optimization of Long-Span Continuum Structures

Herein, to improve the dynamic performance of continuum structures, their fundamental frequency is optimized using the topology optimization method. This helps to obtain the best material distribution in the design space and increases the fundamental frequency of the structure higher than the disturbance frequency. Using the variable density method, the dynamic topology optimization model of a long-span continuum structure is built based on the density interpolation model of a solid isotropic material with penalization (SIMP). The goal of this optimization is to maximize the first-order eigenvalue, and the optimization constraint is that the total volume of the structure is smaller than the given value. To improve the efficiency and accuracy of the model, sensitivity filtering is adopted to avoid numerical instability during calculation. Moreover, the optimization criterion method is used to iteratively solve the optimization results. Finally, the structural topology optimization method is implemented on the long-span single beam of a bridge crane at a construction site. The results show that the natural frequency of the structure is increased and the modal characteristics are improved, which lays the foundation for further optimization and dynamic-response analysis.

Splitting of a photon by an atomic ion

The effect of inelastic splitting of a hard x-ray photon into two photons by an atomic ion, as well as its angular anisotropy, are theoretically investigated for the proposed experimental schemes. These results may be important, in particular, in the diagnostics of magnetic fusion plasmas and in interpreting the background (continuum) structures of emission spectra of hot astrophysical objects.

Alternating optimization of design and stress for stress-constrained topology optimization

Handling stress constraints is an important topic in topology optimization. In this paper, we introduce an interpretation of stresses as optimization variables, leading to an augmented Lagrangian formulation. This formulation takes two sets of optimization variables, i.e., an auxiliary stress variable per element, in addition to a density variable as in conventional density-based approaches. The auxiliary stress is related to the actual stress (i.e., computed by its definition) by an equality constraint. When the equality constraint is strictly satisfied, an upper bound imposed on the auxiliary stress design variable equivalently applies to the actual stress. The equality constraint is incorporated into the objective function as linear and quadratic terms using an augmented Lagrangian form. We further show that this formulation is separable regarding its two sets of variables. This gives rise to an efficient augmented Lagrangian solver known as the alternating direction method of multipliers (ADMM). In each iteration, the density variables, auxiliary stress variables, and Lagrange multipliers are alternatingly updated. The introduction of auxiliary stress variables enlarges the search space. We demonstrate the effectiveness and efficiency of the proposed formulation and solution strategy using simple truss examples and a dozen of continuum structure optimization settings.