Extended optimality in topology design

Most existing studies of 2D problems in structural topology optimization are based on a given (limit on the) volume fraction or some equivalent formulation. The present note looks at simultaneous optimization with respect to both topology and volume fraction, termed here “extended optimality”. It is shown that the optimal volume fraction in such problems — in extreme cases — may be unity or may also tend to zero. The proposed concept is used for explaining certain “quasi-2D” solutions and an extension to 3D systems is also suggested. Finally, the relevance of Voigt’s bound to extended optimality is discussed.     

Graph topology and gap topology for unstable systems

A reformation is provided of the graph topology and the gap topology for a general setting (including lumped linear time-invariant systems and distributed linear time-invariant systems) in the frequency domain. Some essential properties and their comparisons are clearly presented in the reformulation. It is shown that the gap topology is suitable for general systems rather than square systems with unity feedback. It is shown that whenever an unstable plant can be stabilized by feedback, it is a closed operator, mapping a subspace of the input space to the output space. Hence, the gap topology can always be applied whenever the unstable plant can be stabilized. The graph topology and the gap topology are suitable for different subsets of systems and have many similar characteristics. If one confines them to the same subset, they will be identical. The definitions of the graph metric and the gap metric are discussed     

Dynamical systems and topology optimization

This paper uses a dynamical systems approach for studying the material distribution (density or SIMP) formulation of topology optimization of structures. Such an approach means that an ordinary differential equation, such that the objective function is decreasing along a solution trajectory of this equation, is constructed. For stiffness optimization two differential equations with this property are considered. By simple explicit Euler approximations of these equations, together with projection techniques to satisfy box constraints, we obtain different iteration formulas. One of these formulas turns out to be the classical optimality criteria algorithm, which, thus, is receiving a new interpretation and framework. Based on this finding we suggest extensions of the optimality criteria algorithm. A second important feature of the dynamical systems approach, besides the purely algorithmic one, is that it points at a connection between optimization problems and natural evolution problems such as bone remodeling and damage evolution. This connection has been hinted at previously but, in the opinion of the authors, not been clearly stated since the dynamical systems concept was missing. To give an explicit example of an evolution problem that is in this way connected to an optimization problem, we study a model of bone remodeling. Numerical examples, related to both the algorithmic issue and the issue of natural evolution represented as bone remodeling, are presented.     

Linear feedback systems and the graph topology

It is established for general linear systems that the gap metric induces the coarsest topology with respect to which both closed-loop stability and closed-loop performance are robust properties. In earlier works, similar topological results were obtained by exploiting the existence of particular coprime-factor system representations, not known to exist in general. By contrast, the development here does not rely on any specific system representations. Systems are simply characterized as subspaces of norm bounded input-output pairs, and the analysis hinges on the underlying geometric structure of the feedback stabilization problem. Unlike other work developed within such a framework, fundamental issues concerning the causality of feedback interconnections are discussed explicitly. The key result of this paper concerns the difference between linear feedback interconnections, with identical controllers, in terms of a performance/robustness related closed-loop mapping. Upper and lower bounds on the induced norm of this difference are derived, allowing for possibly infinite-dimensional input-output spaces and time-varying behavior. The bounds are both proportional to the gap metric distance between the plants, which clearly demonstrates the gap to be an appropriate measure of the difference between open-loop systems from the perspective of closed-loop behavior. To conclude, an example is presented to show that bounds of the form derived here for linear systems do not hold in a general nonlinear setting     

Structural topology optimization of multibody systems

Flexible multibody dynamics (FMD) has found many applications in control, analysis and design of mechanical systems. FMD together with the theory of structural optimization can be used for designing multibody systems with bodies which are lighter, but stronger. Topology optimization of static structures is an active research topic in structural mechanics. However, the extension to the dynamic case is less investigated as one has to face serious numerical difficulties. One way of extending static structural topology optimization to topology optimization of dynamic flexible multibody system with large rotational and transitional motion is investigated in this paper. The optimization can be performed simultaneously on all flexible bodies. The simulation part of optimization is based on an FEM approach together with modal reduction. The resulting nonlinear differential-algebraic systems are solved with the error controlled integrator IDA (Sundials) wrapped into Python environment by Assimulo (Andersson et al. in Math. Comput. Simul. 116(0):26–43, 2015). A modified formulation of solid isotropic material with penalization (SIMP) method is suggested to avoid numerical instabilities and convergence failures of the optimizer. Sensitivity analysis is central in structural optimization. The sensitivities are approximated to circumvent the expensive calculations. The provided examples show that the method is indeed suitable for optimizing a wide range of multibody systems. Standard SIMP method in structural topology optimization suggests stiffness penalization. To overcome the problem of instabilities and mesh distortion in the dynamic case we consider here additionally element mass penalization.     

A topology for secure MVS systems

Ultimate protection of computers against programming users appears unachievable. True security seems within reach only within systems without programming users. However, programming has to be done within each computing centre. To meet these conflicting ends, this paper proposes a means of isolating any enterprise's vital data from abuse by fully mutually isolating systems at three security levels from one another.The approach proposed is already partly implemented in major computing centres, though with an effectiveness far from that required. Specifically, it is shown that shared DASD degrades the overall security level to that of the least secure system connected. A higher degree of security, as this paper suggests, is reachable in current systems by defining and implementing a three-level (minimal) topology as part of an overall security strategy.     

Pattern generation, topology, and non-holonomic systems

We consider the problem of achieving a desired steady-state effect through periodic behavior for a class of control systems with and without drift. The problem of using periodic behavior to achieve set-point regulation for the control systems with drift is directly related to that of achieving unbounded effect for the corresponding driftless control systems. We prove that in both cases, the ability to use periodic behavior, and more generally, bounded behavior, to achieve the desired goal implies, under a certain topological condition, the non-holonomicity of the control systems. We also prove that under a regularity condition, the resulting system trajectories must be area-generating in a precise sense.     

多智能体系统一致性问题的控制器与拓扑协同优化设计

本文考虑多智能体系统一致性问题的控制与拓扑协同优化设计.首先在给定的二次性能指标下,对多智能体系统的分布式一致性控制协议寻优,得到依赖于网络拓扑图拉普拉斯矩阵的最优控制器.其次,为进一步最大限度地减少拓扑之间的连边,又不降低多智能体系统的收敛速度,通过权衡系统的通信能量和控制能量,寻求网络拓扑的优化设计,给出了拓扑优化算法和多智能体系统特征值的优化方法.最后,仿真研究验证了在控制器优化的基础上进一步寻求拓扑优化,可大大提升系统的一致性性能.     

Unfolding-based diagnosis of systems with an evolving topology

We propose a framework for model-based diagnosis of systems with mobility and variable topologies, modelled as graph transformation systems. Generally speaking, model-based diagnosis is aimed at constructing explanations of observed faulty behaviours on the basis of a given model of the system. Since the number of possible explanations may be huge, we exploit the unfolding as a compact data structure to store them, along the lines of previous work dealing with Petri net models. Given a model of a system and an observation, the explanations can be constructed by unfolding the model constrained by the observation, and then removing incomplete explanations in a pruning phase. The theory is formalised in a general categorical setting: constraining the system by the observation corresponds to taking a product in the chosen category of graph grammars, so that the correctness of the procedure can be proved by using the fact that the unfolding is a right adjoint and thus it preserves products. The theory should hence be easily applicable to a wide class of system models, including graph grammars and Petri nets.     

Controllability of heterogeneous multi-agent systems under directed and weighted topology

This paper considers the controllability problem for both continuous- and discrete-time linear heterogeneous multi-agent systems with directed and weighted communication topology. First, two kinds of neighbour-based control protocols based on the distributed protocol of first-order and second-order multi-agent systems are proposed, under which it is shown that a heterogeneous multi-agent system is controllable if the underlying communication topology is controllable. Then, under special leader selection, the result shows that the controllability of a heterogeneous multi-agent system is solely decided by its communication topology graph. Furthermore, some necessary and/or sufficient conditions are derived for controllability of communication topology from algebraic and graphical perspectives. Finally, simulation examples are presented to demonstrate the effectiveness of the theoretical results.     

Extended Nehari problem for time-variant discrete systems

In this paper a suboptimal solution to the time-variant discrete version of the extended Nehari problem is given. The main tool used to obtain such a solution, with prescribed tolerance, is the so-called anticausal stabilizing solution to the reverse discrete-time Riccati equation.     

Extended invariance principle for nonautonomous switched systems

Stability analysis is developed for nonlinear nonautonomous switched systems, trajectories of which admit, generally speaking, a nonunique continuation on the right. For these systems Krasovskii-LaSalle's invariance principle is extended in such a manner to remain true even in the nonautonomous case. In addition to a nonsmooth Lyapunov function with negative-semidefinite time derivatives along the system trajectories, the extended invariance principle involves a coupled indefinite function to guarantee asymptotic stability of the system in question. As an illustration of the capabilities of this principle, a switched regulator of a fully-actuated manipulator with frictional joints is constructed.     

Multi-physics interpolation for the topology optimization of piezoelectric systems

Unlike single-physics static structural problems, the topology optimization of piezoelectric systems involves three different material coefficient groups, such as the mechanical, electrical, and electromechanical coefficient groups, which correspondingly necessitate three values of penalty exponents. Earlier investigations have shown that stable convergence cannot be ensured if the exponents are selected arbitrarily or based simply on typical rules used for static structural problems. Here, stable convergence implies that the use of more materials can yield better system performance and a distinct void–solid distribution is favored over an intermediate material distribution during the optimization process. However, no rule for choosing the values of penalty exponents has yet been studied for piezoelectric systems and in fact, they have been chosen mainly by trial and error. In this work, two conditions that the three penalty exponents must satisfy for stable convergence are derived for one-dimensional problems and their effectiveness for two-dimensional problems is investigated. The first condition is an intrinsic condition ensuring better energy conversion efficiency between mechanical and electric energy for more piezoelectric material usage and the second one is an objective-dependent condition favoring a distinct void–solid distribution over an intermediate material distribution for the same amount of piezoelectric material used. In this study, we consider the design of piezoelectric actuators, sensors and energy harvesters that can be analyzed by static analysis. Several numerical examples are solved to check the validity of the proposed conditions.     

Controllability of discrete-time multi-agent systems with directed topology and input delay

This paper investigates the controllability of first-order and second-order discrete-time multi-agent systems with directed topology and input delay. The problem is studied in the leader–follower framework where a number of agents are selected to be leaders and serve as control inputs to all other agents. Sufficient and necessary conditions are derived for the controllability of first-order discrete-time multi-agent systems. With sampling period and feedback gain satisfying certain conditions, it is proved under three different protocols that the controllability of second-order discrete-time multi-agent systems is equivalent to that of a pair of submatrices of Laplacian matrix. In addition, the controllability of both first-order and second-order discrete-time multi-agent systems with input delay is shown, through some transformations, to be equivalent to that of the transformed non-delayed systems. Finally, some simulation examples are given to illustrate the theoretical results.     

Virtual topology dynamics and handover mechanisms in Earth-fixed LEO satellite systems

Handling network mobility in a highly dynamic LEO satellite network is a critical issue to achieve seamless and efficient integration of satellite and terrestrial networks. In Earth-fixed satellite systems, this task could be simplified by representing the network with a more static virtual topology. Virtual node (VN) approach is widely explored in satellite networks research; however, it has some deficiencies due to necessity of one-to-one correspondence between virtual nodes and physical satellites. In this work, a generic virtual topology model, namely multi-state virtual network (MSVN) architecture that alleviates these deficiencies, is proposed. A new mathematical model for MSVN is introduced along with its potential contribution to the overall system availability. Furthermore, possible handover mechanisms in Earth-fixed satellite systems are investigated, and new efficient handover mechanisms for both VN-based systems and MSVN-based systems are proposed. Comparing the handover mechanisms, we conclude with the benefits of using an MSVN-based Earth-fixed satellite system.     

非平衡拓扑结构的多智能体网络系统一致性协议

针对多智能体网络系统,本文分别研究了非平衡拓扑结构的多智能体连续状态和离散状态下的一致性协议.首先提出了能使用有向图表示的多智能体网络系统的拓扑结构,并根据该拓扑结构建立了网络系统的1阶数学模型和提出了多智能体网络系统一致收敛准则.对于多智能体网络连续系统,该系统的一致平衡点最终收敛于初始状态的凸组合,本文最终确定了非平衡拓扑结构的一致平衡点.如果多智能体网络系统的拓扑结构没有改变,在离散状态下系统的一致平衡点仍收敛于初始状态的凸组合,并且离散状态下系统的一致平衡点与连续状态下系统的一致平衡点相等.最后采用8个智能体组成的网络拓扑进行计算机仿真,验证理论的正确性.     

多智能体系统通信拓扑最优设计

运用控制理论,矩阵论及最小二乘等理论,研究了多智能体系统的分组一致性与系统通信拓扑图的拉普拉斯矩阵属于特征值0的特征向量之间的关系.给出了在线性协议控制下,系统达到一致性和分组一致性,其通信拓扑的设计方法.提出了一阶多智能体系统的总能量概念,并得到了系统在能量最省时通信拓扑的最优设计.仿真实例佐证本文主要结论的正确性.     

Structural topology synthesis with dynamics and nonlinearities using equivalent linear systems

Topology optimization for nonlinear and dynamic problems is expensive because of the necessity to solve the equations of motion at every optimization iteration in order to evaluate the objective function and constraints. In this work, an iterative methodology is developed using the concept of an equivalent linear system for the topology synthesis of structures undergoing nonlinear and dynamic response, using minimal nonlinear response evaluations. The approach uses equivalent loads obtained from nonlinear dynamic analysis to perform optimization iterations, during the course of which the nonlinear and dynamic system is continuously approximated. In this process, the optimization is decoupled from the nonlinear dynamic analysis. Results are presented for various kinds of nonlinear and dynamic problems showing the effectiveness of the developed approach.     

无结构P2P系统的重叠网拓扑优化

无结构P2P（Peer-to-Peer）系统的自身结构特征表现着良好的自治性和扩展性。然而,由于自身松散的重叠网拓扑结构以及对等节点可以自由地加入和离开的特点,系统十分容易产生重叠层与底层物理网络的拓扑结构不匹配的问题。另一方面,由于无结构P2P系统大多数采用泛洪式转发,大量的消息会通过低效地重叠网连接占用带宽,产生不必要的数据冗余,从而影响网络的性能,降低整个网络的利用率。提出一种动态拓扑优化模型机制,该机制通过节点在消息转发过程中获取实时的网络拓扑信息,进而通过一系列优化策略对低效的拓扑结构实施优化。