Non-parametric shape optimization method for designing cable net structures in form finding and stiffness maximization problems

Pre-tensioned cable net systems are being increasingly used in building façades. Due to the geometrical nonlinearities exhibited by the cables, people have to resort to the nonlinear response history analysis approach for analyzing the seismic responses of cable net structures. This article studies the effectiveness of linear response spectrum analysis method for nonlinear cable net systems. An energy-based parameter is used to describe the structural stiffness, based on which the equivalent single-degree-of-freedom systems for cable net structures with various irregularities could be established. By means of a pseudo-energy-equivalent approach, the nonlinear equivalent single-degree-of-freedom systems are transformed into linear ones, whose seismic responses are reasonable approximations of the accurate nonlinear results. It is revealed that the vibrating period of the equivalent linearized system is close to that of the nonlinear equivalent single-degree-of-freedom system, indicating that the nonlinear effects on seismic responses of cable net structures are not serious. Inspired by the theoretical analysis, large-scale numerical computations are carried out to check the accuracy of linear response spectrum analysis for nonlinear cable net structures. In the numerical computation, four cable net models are excited by 80 seismic records. Considering 19 different degrees of earthquake intensity, a total number of 6080 nonlinear response history analyses are executed, and the results are compared with those obtained from the linear response spectrum analysis. From the numerical results, the linear response spectrum analysis yields proper overestimations of seismic responses with errors within 35%, which may lead to slightly conservative yet acceptable designs in engineering practice. Given all the aforementioned discussions, it is concluded that the linear response spectrum analysis method is effective for seismic response analysis and aseismic design of nonlinear cable net structures.

Active surface adjustment of cable net structures is becoming significant when large-size cable net structures are widely applied in various fields, especially in satellite antennas. A general-duty adjustment method based on active cables is proposed to achieve active surface adjustment or surface profile reconfiguration of cable net structures. Piezoelectric actuators and voice coil actuators are selected for constructing active cable structures and their simplified mechanical models are proposed. A bilevel optimization model of active surface adjustment is proposed based on the nonlinear static model established by the direct stiffness method. A pattern search algorithm combined with the trust region method is developed to solve this optimization problem. Numerical examples of a parabolic cable net reflector are analyzed and different distribution types of active cables are compared.

Wave-based vibration control of large cable net structures

A numerical strategy for relieving pretension of cable net structures by exchanging stiffness components

The active vibration control of cable net structures by piezoelectric stacked actuators is the subject of the present work. The focus of this paper lies on the proposal of a nonlinear method to address the vibration problem for such structures in the presence of uncertainties. To this end, a cable net structure with integrated piezoelectric stacked actuators is considered. For the structure, a nominal model is derived through the finite element method, and an uncertain model is constructed on the basis of the nominal model by taking into consideration the uncertainties. Based on the established uncertain model, a nonlinear state feedback control is deduced through the Lyapunov stability theory. This control guarantees that every uncertain system response enters and remains within a certain neighborhood of the zero state after a finite interval of time. To demonstrate the superiority of the proposed nonlinear control, a linear control based on the linear quadratic regulator is presented for comparison. Displacement responses and control voltages of the uncertain system under an impulse excitation are simulated for both the nonlinear and linear control cases. It is observed that the nonlinear method is feasible and robust as it stabilizes the uncertain system to a smaller size of the region of ultimate boundedness. Besides, the nonlinear approach improves the performance of the uncertain system by reducing the vibration to a lower amplitude within a shorter time.

This work presents dynamic response of large strain deformed subsea flowlines and jumpers conveying two-phase fluid. Large strain deformation theory is used to analyze the effects of high pressure-high temperature two-phase flow on critical velocity in flexible pipes. Starting with natural frequency of simply supported flexible pipes, corresponding nonlinear transport equations are derived and subsequently expanded using the method of multiple scales perturbation. Frequency response obtained from the linearized leading order equations established that buckling and flutter-like instabilities attend critical velocities. Furthermore, the buckling velocity of transverse pipe decreases with increasing temperature, pressure or tension. Clearly, the transverse buckling velocity is higher in large strain deformation model compared with small strain model. In addition; whereas the critical velocity in the longitudinal direction is independent of temperature, pressure and tension for small strain models; these variables determine the path to failure in large strain deformed pipes.Keywords: Large strain deformation, small strain deformation, simply supported pipe

Motivated by the works on non-gradient techniques in the domain of shape optimization of the structure, the present work intends to suggest a novel non-gradient procedure for shape optimization of structures and compare it to an existing gradient-based method. The presented technique optimizes the shape of structural parts using a fuzzy controlled integrated zero-order methodology incorporating the notion of design elements and automated mesh construction with mesh refinement at each iteration. The movement of nodes and convergence monitoring is taken care of using the triangular fuzzy membership function. The changes in shape occur according to the selected target maximum shear stress (σt) with a view of reaching as near to the target as possible at all the points. The present methodology is packaged in a piece of software termed GSO (Gradientless shape optimization) coded in FORTRAN language. To explain the efficacy of the current approach, a few basic structural shapes have been optimized under various constraints, and the results of the same are compared to those obtained using Optistruct (a part of software suite HyperWorks from Altair engineering), which works on gradient descent method. The proposed approach works well and produces more industry fabricable results than what is produced by the gradient descent method in Optistruct.

Cable-net structures are used for many structural purposes, such as stadiums, roofs, bridges…etc. They are lightweight structures that can be used in unique construction at an effective cost. Geometrical nonlinearity governs the performance of cable net systems. This particular system can equilibrate applied loads by undergoing significant deformations with small strains. Therefore, the cable-net structures require to attain a suitable degree of prestressing to prevent cables from slacking and to obtain specific geometry and function. The effective numerical approach is applied for computing the desired level of prestress for a three-dimensional cable-net model and a conical cable-net model. The targeted prestress is achieved considering the nonlinear behavior of cables. The nonlinear member variation is introduced as a second-order function of displaced joints. Then used in determining the desired prestress. Two numerical examples are conducted using the present technique and the nonlinear analysis of SAP2000. Both of the analysis outcomes for the models showed a very well agreement with reaching the target. However, using the Euclidean norm index with a value of 0.0809 in the first example confirmed that the current technique is more approachable to the desired prestress. In addition, when the value of actuation is pre-determined and used in computing the degree of prestress, both the present approach and SAP2000 software work equivalently, as seemed in the second example, which showed 0.04% of the maximum difference in the prestress computation.

A numerical method of structural dynamic shape optimization is presented by using reproducing kernel particle method (RKPM), by which the mesh distortion that exists in shape optimal method based on finite element can be eliminated completely and the optimal model for structural dynamic optimization design is built. The discreteness-based design sensitivity analysis in both natural frequency and dynamic response is proposed by using direct differentiation method and discrete derivatives on the basis of structural dynamic analysis, in which the penalty method is employed into imposing the essential boundary conditions, and the derivatives of shape functions with respect to design variables are derived. The algorithm of dynamic sensitivity analysis is testified by numerical example, and the numerical results obtained are in good agreement with those obtained using semi-analytical method and global finite differences method. Finally, by integrating the algorithm mentioned based on RKPM with parameterized descriptive method of boundary shape, two examples for structural dynamic shape optimization are performed.

This paper is intended to overview on analytical and numerical methods in shape optimization. We compute and analyse the shape Hessian in order to distinguish well‐posed and ill‐posed shape optimization problems. We introduce different discretization techniques of the shape and present existence and convergence results of approximate solutions in case of well posedness. Finally, we survey on the efficient numerical solution of the state equation, including finite and boundary element methods as well as fictitious domain methods. Copyright © 2008 John Wiley & Sons, Ltd.

To meet the demand for light-weight structures, many researchers have investigated TPMS (Triply Periodic Minimal Surface) structures for various micro design applications. This study proposes a multi-scale, non-parametric shape and thickness optimization method for TPMS-based micro-shell structures embedded within a macrostructure. The effective properties of these porous micro-shells are evaluated using the NIAH (New Implementation of Asymptotic Homogenization) method. The distributed thickness and shape of the TPMS micro-shell structures are used as design variables. By minimizing the squared error norm between actual and target displacements at arbitrary macrostructure points, the method controls displacement while satisfying a total volume constraint that includes the micro-shell volume. The optimization problem is formulated as a distributed parameter optimization, with sensitivity functions derived using the numerical derivative and adjoint variable methods. These sensitivities are then integrated into a scalar-type H1 gradient method for thickness optimization and a vector-type H1 gradient method for shape optimization. Numerical examples confirm the method’s effectiveness in determining optimal shape and thickness distributions of TPMS-based micro-shell structures to achieve desired displacement control within a macrostructure.

This chapter is the printed matter of a series of lectures on shape optimization given in Udine (Italy) in June 2012. It is mainly focused on the optimal design problem in elasticity.It is well known, since the pioneer work of Tartar and Murat in the late 70’s, that this problem – as well as the scalar shape optimization problem in conductivity – is ill posed and does not always admit solutions. If the goal is to construct numerical methods to address this optimal design problem, this illposedness will lead to algorithms showing instabilities and erratic behaviours.Two ways are possible to slightly change the initial problem so that it may be solved in an efficient and reliable way by numerical methods. The first one consists in enlarging the set of admissible shapes and allow, as solutions, “generalized shapes” that may contain fine mixtures of the initial material and void. This process is called “relaxation” and the main mathematical tool involved is the homogenization theory. It leads to the so called “homogenization method for shape optimization”. It is described in Section 2, both from theoretical and numerical point of views, including practical issues necessary to write a numerical code.The second approach consists in restricting the set of admissible solutions. A few theorems prove the existence of solutions under further geometrical, topological or regularity constraints. The classical domain variation method is revisited using the level set representation, that may be seen as a constraint on the set of admissible solutions. The main tool is here the shape derivative. The level set algorithm is presented in Section 3.Of course this chapter is too short to develop new results and methods and present extensive proofs. Its goal is rather to give an overview on different parts of our work on this topics. Interested readers should check the references cited to have further details.KeywordsTopology OptimizationShape OptimizationHomogenization MethodAdmissible SolutionOptimal Design ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Because of their superior mechanical, structural, and electronic properties, carbon nanomaterials (CNs) (e.g. graphene sheets and carbon nanotubes) are supposed to be base materials for nanoelectromechanical systems (NEMS). In the present work, we propose a structural optimization method of carbon nanomaterials by introducing topological defects, which consists of the molecular mechanics method, the free-form optimization method, the Phase-Field-Crystal (PFC) method, Voronoi tessellation, and molecular dynamics (MD) simulation. The C-C bonds of CNs are simulated as equivalent continuum beams by a combination of molecular and continuum mechanics, so the atomic structures of CNs can be treated as frame structures. We adopt the free-form optimization method for frames to determine the optimal shapes of CNs in stiffness maximization problem. For obtaining the stable atomic structures of the optimal shapes of CNs, topological defects are introduced in the optimal shapes of CNs using a combination of PFC method, Voronoi tessellation, and MD simulation. The numerical results show that the compliance of CNs can be significantly reduced according to the structural optimization method, which is helpful for designing CNs components in NEMS.

In today's society, with an increasing need for energy worldwide and at the same time a growing awareness of its negative consequences, optimizing and reducing energy consumption has become important for sustainability. The process of optimizing the appliances in large industry and household scale contributes to the minimization of the global energy consumption. In order to increase the efficiency of these appliances, various optimizations are employed, such as the shape optimization. In this thesis, a shape optimization method is used to modify the geometry of an existing heat exchanger. The heat exchanger is used in a domestic boiler and the shape optimization is a crucial step for its fast development. Currently, the shape of the heat exchanger is highly dependent on its manufacturing process. Thus, more complex shapes are investigated and their effects on the heat transfer and pressure drop are evaluated in this thesis. Two aspects need to be investigated for this heat exchanger optimization: the complex internal flow within the heat exchanger, and the performance of the optimization method as well as its resulting shapes. Since the heat exchanger consists of different flow regimes due to its complex geometry, a detailed study on these regimes (ReD = 2000, 100, and 10) is conducted prior to the optimization procedure. Furthermore, the complex geometry of this heat exchanger leads to different models used in this thesis. Therefore this thesis is divided into three parts based on the models: the three-dimensional single cylinder, the two-dimensional cylinder arrays, and the three-dimensional cylinder array. In optimizing the heat exchanger, the shape optimization method is justified as the most suitable method. Using the ANSYS Fluent's adjoint shape optimization method, the pressure drop and the heat transfer of the heat exchanger are optimized. This method has been used in the past for optimizing various geometries, yet to the author's knowledge, no detailed study on the cylinder array has been published. Consequently, a study on the adjoint parameters and its effect on the overall performance of the optimization procedure is of importance. Furthermore, best practices and recommendations for future work regarding the adjoint optimization procedure are elaborated.

After steel wires and steel cables have been discovered for the construction of long span bridges in the middle of the 19th century, it took another century to establish the use of cables for other buildings. In Europe the Olympic Game roof in Munich for the 1972 Olympics was not the first building with cables, but inspired many clients, architects and engineers to cover large spaces with cable structures. In the 1990s many stadium roofs have been designed using cable structures with only 20 – 30 % of the steel weight compared to conventional steel girders. Also for office buildings and skyscrapers stainless steel cables are now used as tension framework in transparent, almost invisible glass curtain walls. Exhibitions halls, airport terminals, railway stations, museums and concert halls are further applications for the extreme lightness and strength of structural cables. Still bridges challenge engineers and downtown locations ask for landmark designs with architecturally pleasing details, without visible threaded rods or other bulky anchorages. Advanced cable structure specialists can engineer and manufacture cable structures without length adjustment within the cable system. Due to the fabrication to very small tolerances in the factory, ready to install components can be supplied to the construction site.