Dynamic Relaxation Method Research Articles

Investigating the numerical stability of dynamic relaxation methods with automatic load-increment schemes to improve snap-back prediction

Investigating the numerical stability of dynamic relaxation methods with automatic load-increment schemes to improve snap-back prediction

Nonlinear Isogeometric Analyses of Instabilities in Thin Elastic Shells Exhibiting Combined Bending and Stretching

Abstract Thin structures like shells are highly susceptible to mechanical instabilities. They can undergo large nonlinear elastic deformations while exhibiting combined bending and stretching modes of deformation. Here, we propose a new displacement-based finite element (FE) formulation based on the isogeometric discretization of Koiter’s nonlinear shell theory and use of the method of dynamic relaxation (DR) to solve equilibrium configurations of problems involving fine-scale and hierarchical wrinkling and buckling. No imperfection seeding is required to trigger instabilities. The use of the NURBS basis provides a rotation-free, conforming, higher-order spatial continuity such that curvatures and membrane strains can be computed directly from interpolating the position vectors of the control points using spatial finite differences. The pseudo-dissipative FE dynamical system is updated through explicit time integration and made scalable for parallel computing using a message passing interface (MPI). The proposed FE method is successfully benchmarked against several numerical and experimental results.

Multi-helix buckling of a slender compression rod constrained by a cylinder

As early as the mid-18th century, Euler proposed a critical load formula for compression rods with multiple half-sine wave modes. However, a critical load formula for compression rods constrained by cylinders with multi-helix buckling modes has not yet been proposed. In this paper, a mechanical model of multi-helix buckling of a slender compression constrained by a cylinder is established. Utilizing the dynamic relaxation method, a finite element solution strategy is presented. Its notable advantage lies in its capability to compute stable configurations of multiple helical buckling, effectively addressing challenges such as contact identification and convergence issues during computation. Furthermore, parameter analysis quantifies the influence of length, diameter, and annular gap on the critical load, while also analyzes the shear force, bending moment, and contact force after buckling. This paper presents formulas for the critical load of multi-helix buckling and the length of the multi-helix buckling modes, which is suitable for various length, diameter, and annular space gap conditions, and partially validates these results against finite element analysis.

Target shape approximation of funicular surfaces by load control using RBF interpolation

Existing methods for form-finding prioritize discovering optimal shapes under design load conditions, often overlooking the designer’s shape preferences. To address this problem, this research relaxes the load condition and develops a user-friendly form-finding tool that encapsulates a new method for obtaining funicular surfaces that pass through user-prescribed points through load control. The method involves generating smooth load distributions via RBF interpolation of nodal loads, which are repeatedly adjusted so that the equilibrium shape is close enough to the target points. For faster convergence, the step size of the load adjustment is automatically determined through a coarse line search, and a warm start is introduced in form-finding with the dynamic relaxation method. Despite uneven load distributions, the method ensures high accuracy in approximating the surface shape to the desired points while preserving funicularity against the load distributions. Additionally, the resulting load distribution gives designers valuable feedback on the specified target points, as the degree of load non-uniformity indicates the deviation from the equilibrium shape for uniformly distributed loads.

Nonlinear bending examination of nanocomposite cylindrical shell exposed to mechanical loads using dynamic relaxation method

In the current context, the nonlinear bending examination of a carbon nanotube reinforced composite (CNTRC) shell exposed to mechanical loading using the first-order shear deformation shell model via the dynamic relaxation (DR) procedure, is perused. The composite shell is assumed to be reinforced in the longitudinal axis. The curved morphology distribution of CNTs is considered to be functionally graded (FG) or uniform along the shell thickness. Mechanical properties of the constituents are gathered based on the modified rule of the mixture. For verification, the present numeric outcomes are compared with available references and ABAQUS finite element package. Finally, the roles of effective items including carbon nanotubes (CNTs) distributions, thickness-to-radius and length-to-radius ratios, boundary conditions, and the volume fraction of CNTs are investigated on the maximum non-dimension deflection.

Optimal self-stress determination for high-accuracy mesh reflectors design considering the pillow distortion

The pillow distortion is a significant barrier to the high-accuracy design of mesh reflector antennas, which is caused by the compatible deformation between the flexible cable network and the metallic mesh, manifested as the pillow-like distortion of the mesh reflecting surface. The pillow effect deviates the mesh surface from the desired paraboloidal shape leading to the surface error. Unfortunately, current researches usually neglect this effect in the mesh reflector design due to its complicated formulation. Focusing on this phenomenon, this paper proposed an optimal self-stress determination method for high-accuracy mesh reflector design. Firstly, a mechanism model of pillow distortion is proposed based on the cable-membrane constitutive models and coupling relations. The integrated equilibrium equation of the mesh reflector is established and solved by the dynamic relaxation (DR) method, which effectively analyses the influence of the pillow distortion on the mesh reflector. Then, considering the pillow distortion, a high-accuracy design method for the mesh reflector is proposed by optimising the geometrical sizes of the cable network and the metallic mesh combining the DR method with optimisation algorithm. Finally, the proposed method is effectively applied to the design of an eighteen-unit mesh reflector antenna, and the feasibility and effectiveness of the proposed method are demonstrated.

Nonlinear dynamics response of porous functionally graded annular plates using modified higher order shear deformation theory

In this investigation, the nonlinear dynamics of porous annular plates that have been functionally graded are examined under varying time-dependent loads. Both simple supported and fully clamped boundary conditions are taken into account. The mechanical properties of the functionally graded plate in its thickness are considered according to the distribution function of the modified law. In addition, porosity, as a stress-relieving property, is also applied throughout the plate thickness by different functions and pore volume fractions. By adopting Hamilton’s principle, the equations of motion are obtained based on the modified higher-order shear deformation plate theory. Then these partial differential equations are solved using the viscous dynamic relaxation method in conjunction with Newmark’s implicit integration method. The present findings are compared and successfully confirmed with those available in the literature. Finally, the effects of some key factors such as porosity distribution type, pore volume fraction, power-law index, loading conditions, and thickness-to-radius ratio on the dynamic behavior of both simple and fully clamped plates have been studied in detail. The study’s findings indicate that porosity pattern X, characterized by a greater prevalence of interconnected pores, exhibited enhanced resistance to deflection. Conversely, pattern O, featuring a more uniformly distributed pore size, effectively mitigated stress in FG porous materials. These observations offer valuable insights for optimizing plate design and elevating its performance.

New actuation planning method for the analysis and design of active tensegrity structures

This paper deals with actuation planning and control for active tensegrity structures. Within actuation planning, a specific actuation in the form of adjustments of cables’ rest lengths is assigned to the resulting structure to follow a prescribed complex trajectory. The methodology is based on the desired trajectory discretisation and following consecutive actuation optimisation, which employs the dynamic relaxation method. Since the actuation planning problem is treated as static, a controller is designed to suppress possible structural vibrations. With regard to the need to test dynamical properties and simulate the structure’s motion, a methodology for automatic creation of tensegrity computational models in Simscape software is developed in the MATLAB scripting environment. Numerical examples of particular tensegrity structures show the whole algorithm’s efficiency.

Experimental Testing of a Tensegrity Simplex: Self-Stress Implementation and Static Loading

A physical model of the simplest three-dimensional tensegrity unit was built at human scale out of aluminum. Self-stress implementation and static loading tests were performed on this model. At each step, accurate measurements were obtained for all nodal positions and element forces. For the prestressing phase, elongations were imposed, via mechanical devices, in different combinations of elements, called prestress scenarios. Experimental results are compared to the theoretical self-stress state obtained by singular value decomposition of the equilibrium matrix and to numerical simulations using the dynamic relaxation method. It is shown that internal forces follow the same linear trend for all prestress scenarios even if the geometry is significantly impacted. Compressive tests were conducted by hanging masses from the top nodes. It is shown that there exists a unique load-displacement relation that follows the infinitesimal mechanism direction for a finite distance, which depends on the self-stress level. The paper provides a detailed overview of the simplex’s structural behavior using both experimental and numerical results while discussing the limitations of the analysis methods explored.

Bifurcations and Stability Analysis of Elastic Slender Structures Using Static Discrete Elastic Rods Method

Abstract Discrete elastic rods (DER) method provides a computationally efficient means of simulating the nonlinear dynamics of one-dimensional slender structures. However, this dynamic-based framework can only provide first-order stable equilibrium configuration when combined with the dynamic relaxation method, while the unstable equilibria and potential critical points (i.e., bifurcation and fold point) cannot be obtained, which are important for understanding the bifurcation and stability landscape of slender bodies. Our approach modifies the existing DER technique from dynamic simulation to a static framework and computes eigenvalues and eigenvectors of the tangential stiffness matrix after each load incremental step for bifurcation and stability analysis. This treatment can capture both stable and unstable equilibrium modes, critical points, and trace solution curves. Three representative types of structures—beams, strips, and gridshells—are used as demonstrations to show the effectiveness of the modified numerical framework, which provides a robust tool for unveiling the bifurcation and multistable behaviors of slender structures.

A new numerical method for elastic-plastic plate analysis: Theoretical-experimental validation and its application in ratcheting phenomenon

In the present paper, two different numerical methods, generalized differential quadrature (GDQ) and dynamic relaxation method (DR) are employed to analyze elastic-plastic plates. These methods are initially evaluated for elastic beams and then they are utilized for elastic-plastic plates analysis. A new method, GDDR (Generalized Differential Dynamic Relaxation), is developed by the combination of two mentioned methods. The proposed method is validated by theoretical analysis for elastic problems, Abaqus simulation and experimental study for elastic-plastic plates. Reduction in the number of required discretizing nodes, lower computational time and decrease in the number of iterations (about 80%) are the advantages of introduced method compared with the conventional ones. Moreover, this method improves the accuracy of solutions, especially in terms of stresses, by increasing the differentiation accuracy of nodes for elastic-plastic analysis. Comparing the errors of proposed GDDR method with conventional DR method shows about 70% reduction in mean squared error. Moreover, this new method is able to solve problems by using only 3 nodes in each direction, while other methods need at least 8.

Effects of prestress implementation on self-stress state in large-scale tensegrity structure

Tensegrity structures are spatial, reticulated, and lightweight structures that are composed of struts and cables. Stability is provided by a self-stress state between tensioned and compressed members obtained by prestress. Although the form finding of tensegrity structures has attracted significant attention, few studies have focused on the structural concept implementation in large-scale systems and prestress implementation in particular. This work investigates the effects of the prestress implementation on the self-stress state in a large-scale tensegrity structure with interconnected struts. The suspended, 6.2-m-tall structure consists of stainless steel cables and carbon-fiber composite tubes. Different uniform and non-uniform prestress and support condition scenarios were applied and the effects on the self-stress state were investigated experimentally by measuring strains in the carbon tubes, stresses in cables, and reaction forces. Numerical modeling was performed using the dynamic relaxation method, whose predicted stress values were compared to experimental data. The results revealed that the cable-by-cable prestress implementation led to lower effective prestress levels than nominally predicted due to a continuous stress redistribution. Non-uniformly applied prestress resulted in a uniform stress state due to the same continuous stress redistribution. Small joint restrictions in rotational angles and small joint friction hindered geometry and stress distribution from changing when the support conditions changed. The importance of extending the analysis of tensegrity structures beyond form finding and sizing is thus demonstrated; prestress implementation and joint design also need attention.

A fast adaptive PD-FEM coupling model for predicting cohesive crack growth

In this study, nonlocal theory of peridynamic (PD) is coupled with finite element method (FEM) to simulate cohesive crack growth in quasi-brittle materials. The adaptive dynamic relaxation method is adopted to implement quasi-static loading conditions. The advantage of peridynamic method in the modeling of crack propagation is taken into consideration to be coupled with that of finite elements which is computationally less expensive and covers various boundary value problems. In this novel method, the whole domain of the problem is initially dominated by FEM solver with coarse mesh grids. Coupling process is then introduced to those regions which have critical conditions according to damage criterion. The capability of the proposed coupling method in the modeling of cohesive crack growth in quasi-brittle materials is demonstrated. In comparison to other conventional PD-FEM coupling methods, the computational cost of the proposed method is considerably improved and total running time of the solution is significantly attenuated. The validity of the method is confirmed through comparing the results with experimental ones.

The dynamic relaxation form finding method aided with advanced recurrent neural network

AbstractHow to establish a self‐equilibrium configuration is vital for further kinematics and dynamics analyses of tensegrity mechanism. In this study, for investigating tensegrity form‐finding problems, a concise and efficient dynamic relaxation‐noise tolerant zeroing neural network (DR‐NTZNN) form‐finding algorithm is established through analysing the physical properties of tensegrity structures. In addition, the non‐linear constrained optimisation problem which transformed from the form‐finding problem is solved by a sequential quadratic programming algorithm. Moreover, the noise may produce in the form‐finding process that includes the round‐off errors which are brought by the approximate matrix and restart point calculating course, disturbance caused by external force and manufacturing error when constructing a tensegrity structure. Hence, for the purpose of suppressing the noise, a noise tolerant zeroing neural network is presented to solve the search direction, which can endow the anti‐noise capability to the form‐finding model and enhance the calculation capability. Besides, the dynamic relaxation method is contributed to seek the nodal coordinates rapidly when the search direction is acquired. The numerical results show the form‐finding model has a huge capability for high‐dimensional free form cable‐strut mechanisms with complicated topology. Eventually, comparing with other existing form‐finding methods, the contrast simulations reveal the excellent anti‐noise performance and calculation capacity of DR‐NTZNN form‐finding algorithm.

Post-buckling loads of a horizontal pipe string with variable length

Horizontal well is one of the main measures to improve oil and gas production capacity. However, with the extension of horizontal section, the risk of pipe string buckling will be increased. In the quasi-static finite element calculation, it is difficult to identify the contact with the cylindrical wellbore due to the jumping changes of the sinusoidal and helical buckling configurations of the pipe string. Therefore, this method is difficult to solve the calculation convergence problem, especially with the increase of horizontal pipe string length and greater flexibility, this problem becomes particularly prominent. We propose a finite element calculation strategy, namely, using the dynamic relaxation method, to transform the static buckling of the horizontal pipe string into a virtual dynamic problem. We put forward the identification methods of sinusoidal and helical post buckling modes, as well as their critical load determination methods. The research results show that the use of large damping in the dynamic relaxation method can significantly restrain the vibration phenomenon of alternating contact and separation between the horizontal pipe string and the cylinder wall, reduce the response time to reach the stable buckling mode, and also can significantly improve the calculation efficiency. The change trend of critical load is the same when the length of horizontal pipe string is extended for a certain length. Within the extended length range, the horizontal pipe string has the same half sine wave or helix number, and the critical loads of sine and helix buckling will gradually decrease with the extension of the string length, and eventually tend to be constant. This trend is more obvious when the number of half sine waves or helixes of horizontal pipe string is more. The dimensionless critical loads of sinusoidal and helical buckling tend to 1.00 and 2 respectively, and the corresponding dimensionless string lengths are 3.73π and 6.21π, respectively. However, other researchers underestimated the effect of length on the critical load of helical buckling. We have verified the critical loads for sinusoidal and helical buckling of short and long pipe strings. The results of this paper are practical for post-buckling analysis of horizontal wells with various lengths of pipe string.

Development of a Time-Domain Simulation Code for Cross-Flow Vortex-Induced Vibrations of a Slender Structure with Current Using the Synchronization Model

In this paper, the numerical analysis method for the cross-flow vortex-induced vibration (CF VIV) analysis based on the proposed procedure for CF VIV analysis of slender structures is developed. In order to consider the changes in the incoming flow according to the static configuration of the slender structures due to the current, the proposed procedure has three stages. A slender structure is modeled as the lumped-mass line, and the dynamic relaxation method known as the numerical technique for a slender structure with large geometric nonlinearity is applied in the static analysis. The comparison studies with a commercial program are carried out to validate the developed code. The vortex-induced force on slender structures is considered with the synchronization model. To verify the developed CF VIV analysis procedure and numerical method for a slender structure, VIV analysis of the tensioned flexible risers under a uniform and shear current is performed. The simulated results of CF RMS displacement show good agreement with the results of the model test It is found that a tensioned riser vibrates with one dominant frequency in resonance with the nth mode, even though multi-frequencies components of the vortex shedding along the riser due to the shear current occurs.

High-accuracy design for mesh antennas considering the metallic mesh

Metallic meshes have great effects on the surface accuracy and tension distribution of mesh antennas, which present considerable challenges to the high-accuracy design of reflectors. Unfortunately, the metallic mesh is usually neglected in current research due to the complex coupling between the metallic mesh and the cable network. This paper presents a novel high-accuracy design method for the mesh reflector considering the effect of the metallic mesh. A cable-membrane coupling model is first proposed to account for the compatible deformation between the cable network and the metallic mesh. By introducing the topology matrix of the cable-membrane structure, the unified equilibrium equation of the mesh antenna is established and solved by the dynamic relaxation (DR) method. A novel high-accuracy design method for the mesh antenna is proposed by optimising the geometric dimensions of the cables and the membranes. Finally, this method is effectively applied to the high-accuracy design of two mesh antennas with different topologies considering the effect of the metallic mesh. The superiority of the method is demonstrated through a comparison with the literature, and the accuracy of the results is exhibited by the FEM software ABAQUS.

A combination of genetic algorithm and dynamic relaxation method for practical form-finding of tensegrity structures

A combination of Genetic algorithm (GA) and Dynamic Relaxation (DR) method is presented for practical form-finding of tensegrity structures. Several approaches of dynamic relaxation method are reviewed for form-finding of tensegrity structures, and the negative points of DR method where the final form is sensitive to the initial form and cannot satisfy the geometrical constraint conditions are illustrated. The genetic algorithm is then proposed to obtain the initial configuration for DR method for finding the final shapes satisfying all constraint conditions. Several types of tensegrity structures are performed to point out the efficiency of proposed combination method.

A discrete differential geometry-based numerical framework for extensible ribbons

As a typical mechanical structure, ribbons are characterized with three distinctly different dimensions, i.e., length ≫ width ≫ thickness, which leads to their exclusive behaviors different from the one-dimensional (1D) case of slender rods and the two-dimensional (2D) case of thin plates. In this paper, we report a discrete differential geometry (DDG)-based numerical method to simulate the geometrically nonlinear deformations of extensible ribbons. With a cross section-dependent regularized parameter introduced, the proposed 1D framework allows both in-plane stretching and out-of-plane bending in the ribbon mid-surface, aimed at bridging the gap between the linear Kirchhoff rod theory and the developable Sadowsky ribbon model. Instead of solving the ordinary differential equations (ODEs) directly with the associated boundary conditions, the mechanical object is discretized into a mass–spring system, and its equilibrium configuration is obtained through a dynamic relaxation method. The numerical framework is applied to seven typical occasions based on either experimental or published datasets in order to verify its performance. Quantitative agreements demonstrate the effectiveness and accuracy of the proposed discrete approach for extensible ribbons, which, as a computationally efficient numerical simulator, could provide a pivotal understanding of a batch of slender structures, and further inspire the simulation-guided design of man-made systems.

Wind tunnel tests of an inflated membrane structure. Two study cases: with and without end-walls

The use of textile membranes in architecture and civil engineering has increased rapidly in the last decades. Membrane structures are lightweight, economic structures with unique shapes. At the same time, their structural analysis is particular compared to the analysis of conventional structures. The form-finding procedure, the large displacements of the structures, and the special properties of the composite material all require unique tools. The wind analysis of membrane structures is one of the most challenging parts of the design because the design codes do not provide the pressure coefficients of the doubly curved shapes of membrane structures. The main scope of the present research was to determine the mean pressure coefficient fields over an inflated membrane structure by wind tunnel experiments. The structure, composed of six inflated circular arches, was analyzed with and without end-walls for three wind directions. The equilibrium shape of the inflated structure was determined with the Dynamic Relaxation Method. During the numerical form-finding procedure, the orthotropic behavior of the membrane material and the warp and fill directions of the textile fibers were also considered. Based on the numerically determined equilibrium shape of the inflated structure, a model was made using a 3D printer. The flow around the model according to three wind directions was analyzed in an open-circuit wind tunnel. The pressures were measured in 102 external and 102 internal points of the inflated arches and 27 points on each end-wall. In this paper, the experimentally determined pressure coefficient fields for different wind directions are presented and compared for the open (without end-walls) and closed (with endwalls) cases. The thoroughly introduced experimental results can be used during the structural analysis of future inflated structures with similar shapes.