Form Finding Problems Research Articles

Potential Energy Optimization Approach in the Form-Finding of Tensegrities and Cable–Strut Systems

A form-finding process for tensegrities and cable–strut systems is determined as a minimum problem of potential energy that is constructed considering initial estimated pre-stress states. Both the Jacobian and Hessian matrices of a potential energy function are derived explicitly regarding free nodal coordinates. Based on these derivations, both the first-order geodesic dynamic relaxation optimization method and the second-order optimization method of Newton can be applied to solve form-finding problems involving tensegrities and cable–strut structures. It should be noted that both the pre-stress state and the configuration of the systems are adjusted during the optimization process to satisfy self-equilibrium conditions. The geometrical constraints are then included in the proposed work as equality constraint conditions to find the final forms satisfying not only the self-equilibrium states but also the designers’ demands. Two well-known and complex systems are investigated to assess the efficiency of the proposed method, i.e. free-standing truncated icosahedral tensegrity and the Kiewitt Dome.

An improved dog-leg method for form-finding of tensegrity structures

This paper presents a novel and highly efficient form-finding method for tensegrity structures. The method is based on converting the nonlinear equilibrium equations with the nodal coordinates vector as the variables into nonlinear least-squares problems. Subsequently, the Dog-Leg method is applied to solve these transformed problems. Furthermore, the Dog-Leg method is combined with the MBFGS (Modified Broyden-Fletcher-Goldfarb-Shanno) method to overcome the singularity problem of the stiffness matrix. The resulting improved Dog-Leg method, i.e., DL-MBFGS method, can effectively address form-finding problems not only in self-stressed and non-self-stressed tensegrity structures, but also for tensegrities subjected to external loads and other scenarios including mechanisms and large displacements problems, which shows its great versatility and robustness. Several numerical examples are presented to validate the accuracy and effectiveness of the proposed method.

A zeroing neural network model for form-finding problems: A nonlinear optimization approach

How to establish a simple and efficient nonlinear optimization problem form-finding model in noise environments is the main concern of this paper, thereby, a sequential quadratic programming form-finding (SQP) approach via a noise-tolerant zeroing neural network (NTZNN) is presented to solve the proposed problem. First, the rank of force density matrix is selected as optimization objective due to its crucial physical property. In addition, linear relations between the force density vectors and force density matrix are selected as the constraint conditions, which can ensure the tensegrity mechanism in a stable status. Furthermore, the simulations are shown that the presented SQP-NTZNN algorithm has superiority with anti-noise performance and computation efficiency, which by comparing the computational rate in the same noise environment with the novel form-finding algorithm. Eventually, the form-finding simulations under different initial conditions are formed to verify the importance of the constrained conditions, it can prove the high efficiency of the proposed nonlinear constrained optimization model which is transformed from the form-finding problem.

Constrained Form-Finding of Tension–Compression Structures using Automatic Differentiation

This paper proposes a computational approach to form-find pin-jointed bar structures subjected to combinations of tension and compression forces. The generated equilibrium states can meet additional structural and geometrical constraints via gradient-based optimization. We achieve this by extending the combinatorial equilibrium modeling (CEM) framework in three important ways. First, we introduce a new topological object, the auxiliary trail, to expand the range of structures that can be form-found with the framework. Then, we leverage automatic differentiation (AD) to obtain an exact value of the gradient of the sequential and iterative calculations of the CEM form-finding algorithm, instead of a numerical approximation. Finally, we encapsulate our research developments into an open-source design tool written in Python that is usable across different CAD platforms and operating systems. After studying four different structures –a self-stressed tensegrity, a tree canopy, a curved bridge, and a spiral staircase– we demonstrate that our approach enables the solution of constrained form-finding problems on a diverse range of structures more efficiently than in previous work. • A computational form-finding method is extended to enhance the form generation of spatial structures in static equilibrium subjected to tension and compression forces. The output forms can satisfy combinations of force and geometric constraints via gradient-based optimization. • A new topological modeling object, the auxiliary trail, is presented which simplifies the representation of a structure as a valid input to the method herein extended. • Automatic differentiation (AD) is leveraged to obtain an exact value of the gradient of a sequential and iterative form-finding algorithm with marginal human intervention. The application of AD leads to significant computational performance gains, especially when solving large constrained form-finding problems. • A standalone, open-source design tool is developed which enables the formulation and the solution of constrained form-finding problems in plain and simple Python code. The design tool is readily usable on three different operating systems and on three distinct 3D modeling software environments.

A unifying framework for form-finding and topology-finding of tensegrity structures

This paper presents a unifying framework for the form-finding and topology-finding of tensegrity structures. The novel computational framework is based on rank-constrained linear matrix inequalities. For form-finding, given the topology (i.e., member connectivities), the determination of the member force densities is formulated into a linear matrix inequality (LMI) problem with a constraint on the rank of the force density matrix. The positive semi-definiteness and rank deficiency condition of the force density matrix are well managed by the rank-constrained LMI-based formulation. A Newton-like algorithm is employed to solve the rank-constrained LMI problem. Two methods, named direct method and indirect method, are proposed to determine the nodal coordinates once the force densities have been obtained. For topology-finding, given the geometry (i.e., nodal coordinates), the determination of the topology is also formulated into an LMI problem with a constraint on the rank of the tangent stiffness matrix. Numerical examples demonstrate that different types of form-finding problems (such as tensegrity structures with single and with multiple self-stress states, symmetric and irregular tensegrity structures) can be uniformly and efficiently solved by the proposed approach. Furthermore, three well-known tensegrity structures are reproduced to verify the effectiveness of the proposed formulation on the topology-finding of tensegrity structures.

Form-finding method for the target configuration under dead load of a new type of spatial self-anchored hybrid cable-stayed suspension bridges

General form-finding problems of cable-supported bridges are established based on a design scenario in which rigidly fixed starting control points must be given as necessary design constraints prior to independent analysis of any of its cable subsystems. This article presents a form-finding method to address a new case, in which the starting control point serves as an intermediate, flexibly variable connection, to couple two related cable subsystems in a multi-nonlinear environment for the target configuration under dead load (TCUD) of a novel type of spatial self-anchored hybrid cable-stayed suspension (HCSS) bridge. A two-layer framework is proposed by integrating finite element analysis (FEA) and analytical formulas with optimization algorithms to form a self-regulated interactive analysis among subsystems in an iterative manner. The outer layer seeks to achieve self-equilibrium of the global system under the control information of the TCUD, while the inner layer optimizes the subsystems in terms of the initial tensions in the main cables, stay-cables, branches and hangers to obtain a rational mechanical state of the bridge. Then, the TCUD and the intermediate starting control points are determined. To achieve computational stability, a high-performance accelerated Steffens-Newton (ASN) differential algorithm is developed for the shape finding of the cable-hanger subsystem, whereas the non-dominated sorting genetic algorithm (NSGA-II) is adopted as a multiobjective optimizer for TCUD optimization of the other subsystems. The proposed framework is applied to a real-scale self-anchored HCSS bridge, and its validity and performance are demonstrated by comparison studies with a non-optimal scheme and in-field test data.

Innovative numerical form-finding of tensegrity structures

An innovative numerical platform for the form-finding of tensegrity structures is proposed. A particular objective function is introduced based on LU-decomposition of the force density matrix. The required rank deficiency of the force density matrix, which is a crucial quantity in the form-finding process, is then pursued by minimisation of the mentioned objective function. The gradient and hessian of the objective function are also derived explicitly. This critical feature enables us to efficiently employ powerful nonlinear programming approaches in the form-finding problems. Four numerical examples are provided to validate the feasibility and effectiveness of the method proposed. The results compared and contrasted from different aspects with existing analytical and some well-known numerical solutions. The comparisons verify that the method proposed is highly accurate and fast; therefore, it can be used as a compelling alternative to the available numerical form-finding methods.

A Task-Space Form-Finding Algorithm for Tensegrity Robots

A modified dynamic relaxation algorithm for the form-finding of tensegrity robots in task space is proposed in this paper. There are some form-finding problems of tensegrity structures that are hard to be handled by the node-based dynamic relaxation algorithm. For tensegrity structures with multiple interconnected rods or rigid rods, the node-based dynamic relaxation algorithm is usually difficult to perform this type of form-finding. The node-based dynamic relaxation algorithm becomes rather cumbersome if several different ratios between rod lengths and cable lengths are desired, which restricts its applicability to less regular structural forms. The performance of the algorithm is affected by the relative difference in stiffness between members(cables and rods). To solve these problems, we extend the general dynamic relaxation algorithm from node space to task space. The proposed algorithm considers multiple interconnected rigid rods as an entire rigid body to derive its motion in six degrees of freedom. We simulated the motion process of the rigid body approaching its self-equilibrium position under the force exerted by all surrounding cables. In the form-finding process, the relative location relationship of all interconnected rods on a complex rigid body is maintained. The algorithm is valid for the form-finding of tensegrity structures which are with several different ratios between rod lengths and cable lengths. Besides, the modified algorithm is not affected by the stiffness of interconnected rods on rigid bodies. To demonstrate the capability of the method, four examples, including a two-segment spine robot, a bio-inspired tensegrity arm, a multi-segment spine robot, a tensegrity robot with disconnected and interconnected rigid rods, are presented in this paper.

Evolutive and Kinematic Limit Analysis Algorithms for Large-Scale 3D Truss-Frame Structures: Comparison Application to Historic Iron Bridge Arch

Two new computational algorithms for the Limit Analysis (LA) of large-scale 3D truss-frame structures recently proposed by the authors are reconsidered and adapted for a comparison prediction of the elastoplastic response of a strategic beautiful historic infrastructure, namely the Paderno d’Adda bridge (or San Michele bridge), a riveted wrought iron railway viaduct that was built in northern Italy in 1889. The first LA algorithm traces a fully exact evolutive piece-wise linear elastoplastic response of the structure, up to plastic collapse, by reconstructing the true sequence of activation of made-available plastic joints (as a generalization of plastic hinges), in the true spirit of LA. The second LA algorithm develops an independent kinematic iterative approach apt to directly determine the plastic collapse state, in terms of collapse load multiplier and plastic mechanism, based on the upper-bound theorem of LA. Specifically, the marvelous doubly built-in parabolic arch of the bridge is analyzed, under a static loading configuration at try-out stage, and its elastoplastic response is investigated, in terms of evolutive load-displacement curve, collapse load multiplier and plastic collapse mechanism. The two LA algorithms are found to much effectively run and perform, despite the rather large size of the computational model, with a number of dofs in the order of four thousands, by achieving good corresponding matches in terms of the estimate of the load-bearing capacity and of the collapse characteristics of the arch substructure, showing this to constitute a well-set structural element. Moreover, the direct kinematic method displays a rather dramatic performance, in truly precipitating from above onto the collapse load multiplier and rapidly adjusting to the collapse mode, in very few iterations, by a considerable saving of computational time, with respect to the complete evolutive elastoplastic analysis. This shall open up the way for further adoption of such advanced LA tools, with LA regaining a new momentum within the modern optimization analysis of structural design and form-finding problems.

Mechanical models in computational form finding of bending-active structures

This article reviews the different aspects involved in computational form finding of bending-active structures based on the dynamic relaxation technique. Dynamic relaxation has been applied to form-finding problems of bending-active structures in a number of references. Due to the complex nature of large spatial deformations of flexible beams, the implementation of suitable mechanical beam models in the dynamic relaxation algorithm is a non-trivial task. Type of discretization and underlying beam theory have been identified as key aspects for numerical implementations. References can be classified into two groups depending on the selected discretization: finite-difference-like and finite-element-like. The first group includes 3- and 4-degree-of-freedom implementations based on increasingly complex beam models. The second gathers 6-degree-of-freedom discretizations based on co-rotational three-dimensional Kirchhoff–Love beam elements and geometrically exact Reissner–Simo beam elements. After reviewing and comparing implementation details, the advantages and drawbacks of each group have been discussed, and open aspects for future work have been pointed out.

Pity beetle algorithm – A new metaheuristic inspired by the behavior of bark beetles

In the past years a great variety of nature-inspired algorithms have proven their ability to efficiently handle combinatorial optimization problems ranging from design and form finding problems to mainstream economic theory and medical diagnosis. In this study, a new metaheuristic algorithm called Pity Beetle Algorithm (PBA) is presented and its efficiency against state-of-the-art algorithms is assessed. The proposed algorithm was inspired by the aggregation behavior, searching for nest and food, of the beetle named Pityogenes chalcographus, also known as six-toothed spruce bark beetle. This beetle has the ability to locate and harvest on the bark of weakened trees into a forest, while when its population exceeds a specific threshold it can infest healthy and robust trees as well. As it was proved in this study, PBA can be applied to NP-hard optimization problems regardless of the scale, since PBA has the ability to search for possible solutions into large spaces and to find the global optimum solution overcoming local optima. In this work, PBA was applied to well-known benchmark uni-modal and multi-modal, separable and non-separable unconstrained test functions while it was also compared to other well established metaheuristic algorithms implementing also the CEC 2014 benchmark and complexity evaluation tests.

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

Cable net structures, belonging to the tension structures, have been widely utilized in structural engineering. In this study, we develop a non-parametric shape optimization method based on the H1 gradient method for designing cable net structures in form finding and stiffness maximization problems. We divide the design velocity field of cable net structures into the off-axis and the in-axis components for deriving and applying the shape gradient function in each design problem conveniently. In the form finding problem, we use the total length of cables as the objective functional and minimize it without or with considering the constraint condition of perimeter. In the stiffness maximization problem, the compliance is used as the objective functional and minimized under the constraint condition of perimeter and total length of cables considering large deformation and small strain. Moreover, two-step shape optimization of cable net structures from the form finding problem to the stiffness maximization problem is also implemented. The validity of the developed shape optimization method is confirmed by design examples.

Application of a Non-Invasive Form Finding Algorithm to the Ring Compression Test with Varying Friction Coefficients

It is a great challenge in the development of functional components to determine the optimal blank design (material configuration) of a workpiece according to a specific forming process, while knowing the desired target geometry (spatial configuration). A new iterative non-invasive algorithm, which is purely based on geometrical considerations, is developed to solve inverse form finding problems. The update-step is performed by mapping the nodal spatial difference vector, between the computed spatial coordinates and the desired spatial target coordinates, with a smoothed deformation gradient to the discretized material configuration. The iterative optimization approach can be easily coupled non-invasively via subroutines to arbitrary finite element codes such that the pre-processing, the solving and the post-processing can be performed by the habitual simulation software. This is exemplary demonstrated by an interacting between Matlab (update procedure for inverse form finding) and MSC.MarcMentat (metal forming simulation). The algorithm succeeds for a parameter study of a ring compression test within nearly linear convergence rates, despite highly deformed elements and tangential contact with varying friction parameters.

Tangential differential calculus and the finite element modeling of a large deformation elastic membrane problem

We develop a finite element method for a large deformation membrane elasticity problem on meshed curved surfaces using a tangential differential calculus approach that avoids the use of classical differential geometric methods. The method is also applied to form finding problems.

A comparison between a recursive method based on an inverse mechanical formulation and shape optimization for solving inverse form finding problems in isotropic elastoplasticity

AbstractIn this paper we compare two methods, a recursive method based on an inverse mechanical formulation and a method based on a recursive shape optimization formulation, in order to solve inverse form finding problems in isotropic elastoplasticity. Both methods are succinctly presented and a numerical example is given. It was found that no difference could be found between the node coordinates on the undeformed configurations computed with both methods. However the convergence to the solution is faster with the recursive method based on an inverse mechanical formulation than with the method based on shape optimization. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On a recursive formulation for solving inverse form finding problems in isotropic elastoplasticity

BackgroundInverse form finding methods allow conceiving the design of functional components in less time and at lower costs than with direct experiments. The deformed configuration of the functional component, the applied forces and boundary conditions are given and the undeformed configuration of this component is sought.MethodsIn this paper we present a new recursive formulation for solving inverse form finding problems for isotropic elastoplastic materials, based on an inverse mechanical formulation written in the logarithmic strain space. First, the inverse mechanical formulation is applied to the target deformed configuration of the workpiece with the set of internal variables set to zero. Subsequently a direct mechanical formulation is performed on the resulting undeformed configuration, which will capture the path-dependency in elastoplasticity. The so obtained deformed configuration is furthermore compared with the target deformed configuration of the component. If the difference is negligible, the wanted undeformed configuration of the functional component is obtained. Otherwise the computation of the inverse mechanical formulation is started again with the target deformed configuration and the current state of internal variables obtained at the end of the computed direct formulation. This process is continued until convergence is reached.ResultsIn our three numerical examples in isotropic elastoplasticity, the convergence was reached after five, six and nine iterations, respectively, when the set of internal variables is initialised to zero at the beginning of the computation. It was also found that when the initial set of internal variables is initialised to zero at the beginning of the computation the convergence was reached after less iterations and less computational time than with other values. Different starting values for the set of internal variables have no influence on the obtained undeformed configuration, if convergence can be achieved.ConclusionsWith the presented recursive formulation we are able to find an appropriate undeformed configuration for isotropic elastoplastic materials, when only the deformed configuration, the applied forces and boundary conditions are given. An initial homogeneous set of internal variables equal to zero should be considered for such problems.

Flexible bridge decks suspended by cable nets. A constrained form finding approach

The initial geometry of structures made of cables is steered by the cable tensioning forces. In a cable net the geometrical shape and the internal force distribution cannot be dealt as separate issues: the set of geometries defines also the feasible sets of the internal forces. During the last decades, many different approaches have been proposed to deal with the form finding of cable structures. The most efficient one is the so called Force Density Method (FDM), proposed by Schek, which allows to conforming cable nets for structural applications without requiring any further assumption, neither on the geometry, nor on the material properties. An Extension of the Force Density Method, the EFDM, makes it possible to set conditions in terms of fixed nodal reactions or, in other words, to fix the position of a certain number of nodes and, at the same time, to impose the intensity of the reaction forces. Through such an extension the EFDM enables us to deal with form finding problems of cable nets subjected to given constraints and in particular to treat mixed structures, made of cables and struts. In this paper we consider cable nets interacting with members having flexural behaviour. For a given cable assembly and for a given loading condition, aim of this work is to find that particular pretensioning system which replaces both the static and the kinematic functions of the inner reactions of a flexural elastic continuous beam. It is, for instance, the case of the bridge decks suspended by cables, shaped in various forms. The specialization of the EFDM to this type of problem is presented and a progressive set of examples shows the efficiency and the versatility of this approach in contributing to the design of new creative forms.

Evolutionary development of tensegrity structures

Contributions from the emerging fields of molecular genetics and evo-devo (evolutionary developmental biology) are greatly benefiting the field of evolutionary computation, initiating a promise of renewal in the traditional methodology. While direct encoding has constituted a dominant paradigm, indirect ways to encode the solutions have been reported, yet little attention has been paid to the benefits of the proposed methods to real problems. In this work, we study the biological properties that emerge by means of using indirect encodings in the context of form-finding problems. A novel indirect encoding model for artificial development has been defined and applied to an engineering structural-design problem, specifically to the discovery of tensegrity structures. This model has been compared with a direct encoding scheme. While the direct encoding performs similarly well to the proposed method, indirect-based results typically outperform the direct-based results in aspects not directly linked to the nature of the problem itself, but to the emergence of properties found in biological organisms, like organicity, generalization capacity, or modularity aspects which are highly valuable in engineering.

Design of nexorades or reciprocal frame systems with the dynamic relaxation method

The objective of this paper is to propose a simple method using dynamic relaxation algorithm to find the form and perform a structural analysis of reciprocal frame systems, also called nexorades. After a brief historical note, details of the parameters that govern the design of nexorades are introduced together with details of the assembly. These parameters induce strong geometrical constrains on this kind of structures so that a form-finding step is required. It is suggested here to introduce a fictitious mechanical behaviour to solve this problem. The dynamic relaxation algorithm is used with a model that takes into account the eccentricity between the elements. Its implementation is explained and its versatility is illustrated through several examples covering various fields of applications going from form-finding problems to non-linear structural analysis of structures. In a last example the structural behaviour of nexorades is compared with more conventional triangulated structures.

Proposal to Roof the Courtyards of an Historical Building in Pisa with Glass and Steel Grid Shells: Form Finding and Stability Problems

The technical note describes a proposal to roof two internal courtyards of the main building of the Engineering Faculty at the University of Pisa by means of thin flat shells made of a quadrilateral mesh of steel bars supporting plane glass plates stiffened by a grid of steel cables. An exposure of the structural design concepts and of the geometrical genesis of the shells is presented accompanied by a stability analysis and a description of the technological solutions found to solve some crucial constructional problems.