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Incorporating the arc-length constraint, the dynamic relaxation strategy has been widely used to trace full equilibrium path in the post-buckling analysis of structures. This combined numerical scheme has been shown to be successful for solving snap-through problems, but its applicability to snap-back problems has been rarely investigated and remains unclear. This paper proposes a direct and more general finite-difference equation to investigate the numerical stability of this combined numerical scheme, which is dominated by the spectral radius of amplification matrix. And a key discovery of this paper is that a first minor of the tangent stiffness matrix is always negative once snap back occurs. Due to this negative minor stiffness, the spectral radius is invariably greater than one, resulting in unconditional instability, which demonstrates the invalidity of dynamic relaxation arc-length method for snap-back problems. These important conclusions are corroborated by the numerical results of three representative examples in one-, two- and three-dimensional spaces.

In future infrared photodetectors, single-walled carbon nanotubes (SWCNTs) are considered as potential candidates due to their band gap, high absorption coefficient (104 - 105 cm −1), high charge carrier mobility and ease of processability. The SWCNT based Field Effect Transistors (CNFETs) are being seriously considered for applications in optoelectronics. In the proposed work optically controlled back gated CNFET is modeled in Sentaurus TCAD to observe the impact of high dielectric oxides on its photoabsorption. The model is based on analytical approximations and parameters extracted from quantum mechanical simulations of the device and depending on the nanotube diameter and the different gate oxide materials. A small deviation in SWCNT chirality shows significant change (more than 50 %) in channel current. Transfer characteristics of the device are analyzed under dark and illuminated conditions. CNFET integrated with HfO2 dielectrics exhibits superior performance with a significant rise in photocurrent current. Precise two dimensional TCAD simulation results and visual figures affirm that the ON state performance of CNFET has significant dependency on the dielectric strength as well as width of the gate oxide and its application in enhancing the performance of carbon nanotube based infrared photo detectors.

Dynamic Relaxation Method (DRM) is an explicit approach for solving the simultaneous systems of equations. In this tactic, the fictitious mass and damping are added to the static governing equations, and an artificial dynamic system is constructed. By using DRM for nonlinear analysis, the structural static equilibrium path is obtained. This outcome is extremely valuable, since it leads to the behavior of structures. Among the finding related to the structural static path are the critical and buckling points for nonlinear structures. In this paper, a new way for calculating the load factor is proposed by setting the external work zero. Mixing the dynamic relaxation scheme with external work technique has not been formulated so far. In all incremental-iterative methods, the load factor increment sign should be determinated by extra calculations. This sign leads to increase or decrease of the load increment. It is worth emphasizing that sign of the load factor increment changes at the load limit points. Therefore, the sign determinations are required in the common work control methods. These disadvantages are eliminated in the proposed algorithm. In fact, the suggested load factor depends only on the Dynamic Relaxation (DR) fictitious parameters. Besides, all DR calculations are performed via vector operation. Moreover, the load factor is calculated only by one formula, and it has the same relation in the all solution processes. In contrast to the arc length techniques, which requires the sign determined, the proposed scheme does not need any sign finding. It is shown that author’s technique is quicker than the other dynamic relaxation strategies. To prove the capability and efficiency of the presented scheme, several numerical tests are performed. The results indicate that the suggested approach can trace the complex structural static paths, even in the snap-back and snap-through parts.

The aim of this article is to assess the speed of convergence of numerical calculation of cable-membrane structures using dynamic relaxation method in the process of finding critical damping pre-calculated on undamped system. The procedure is tested on four different constructions in six numerical cases. The variety of examples is as large as possible to demonstrate the greatest versatility of the test procedure. The efficiency of the procedure is evaluated based on the number of iterations.

Common dynamic relaxation (DR) methods use central finite differences to derive fundamental equations. In this study, Lagrangian interpolation functions are utilized to formulate iterative DR relationships with kinetic damping. This procedure leads to an algorithm that, unlike the ubiquitous DR methods, does not require the calculation of nodal velocities thereby marching forward only through successive nodal displacement. Also, the power iteration method is used to determine the optimal time-step ratio. By utilizing this time-step, the restarting analysis phase which is one of the drawbacks of the kinetic DR strategy is eliminated. Furthermore, a new variable load factor is developed based on the concept of minimization of displacement increment function. To evaluate the performance and efficiency of the proposed method, some truss and frame structures with geometrically nonlinear behavior are analyzed. Results prove that the number of convergence iterations and analysis time are reduced in the new kinetic DR scheme in comparison with other conventional DR methods.

The traditional Kinetic Dynamic Relaxation (KDR) method can only deal with simple constraints such as the fixed joints, which restricts its practical applications. This study has proposed a new algorithm to implement the complex position constraints in KDR. The proposed algorithm is developed by transforming the position constraints to the acceleration form through combining time differentiation and Taylor expansion with dimensional analysis, and then solving the governing equation of the constraint forces with the Newton’s 2nd law. For the nonlinear constraints, projection technique is applied to avoid the drift-off phenomenon. Several tests have been performed to verify the proposed algorithm. The numerical results show the algorithm is adaptive in both linear and nonlinear problems and works efficiently.

In this study, a new axisymmetric ordinary state-based peridynamic (PD) model for axisymmetric problems of linear elastic solids is presented. A fracture criterion based on the PD bond energy density is proposed. Adaptive dynamic relaxation (ADR) method is adopted to obtain equilibrium solutions, and a viable fictitious density of the model is derived and proven to be valid for implementation in the ADR method. Performance and validity of the proposed axisymmetric PD model are demonstrated by three kinds of numerical problems, i.e., compression tests, pull-out deformation, and indentation fracture. In the compression tests with focus on constant-strain deformations, the displacements predicted by the present model are compared with classical analytical solutions, and the results show good agreements. Both m- convergence and δ -convergence behaviors under four influence functions are investigated and discussed based on a thorough error analysis in different compression cases. The recovery of the Poisson’s ratios in the model is tested in detail as well. The capability of capturing general non-uniform axisymmetric deformation by the proposed model is verified in the pull-out analysis. The equilibrium displacement fields predicted by the present model agree very well with those by the finite element method. The peridynamic evaluation of strains and stresses also show good match with the finite element ones. The proposed fracture criterion is validated by the third example of indentation cracking which is compared with the available experimental data. The model developed can effectively be used to analyze axisymmetric problems of linear elastic solids in the framework of the ordinary state-based peridynamics.

Dynamic relaxation (DR) is a widely used numerical method to determine the static equilibrium of a dynamic system. However, it is difficult to apply conventional DR methods to nonlinear models because they require estimation of a stiffness matrix of the model. To resolve the forementioned problem, a new dynamic relaxation method using continuous kinetic damping (CKDR) was proposed in previous research. The CKDR method does not require any model parameters including the stiffness matrix, and it possesses absolute stability and a second-order convergence rate. However, the convergence rate is proportional to square of the step size, and this may result in a low convergence rate if the selected step size is excessively small. This problem leads to difficulties in the practical use of CKDR. Thus, an adaptive step-size method is proposed in this paper to control the convergence rate of CKDR. The proposed method estimates natural frequency of the model and determines adaptive step size. Static equilibrium simulations were performed for three different models to verify the method. The results revealed that the computational cost of CKDR with a variable step size was very efficient when compared to fixed step sizes and that the convergence rate was also controlled as intended. In addition, the lowest natural frequencies of models in static equilibrium were accurately estimated.

In this paper, a novel numerical form-finding method is presented based on a dynamic relaxation algorithm for cable nets in mesh reflectors. To obtain a shape with high profile efficiency, structural forces are assumed to be constant values during the computational iterations, so a perfectly uniform distribution of structural forces can be achieved. An initial shape, normally not in equilibrium, is given in advance, and nodes of the network are forced to vibrate by the unbalanced forces. Parameters of dynamic relaxation algorithms, such as the type of damping and the time interval, are tested to ensure the speed and stability of the computation process. Finally, two different types of Astromesh reflectors are used as examples to verify the developed method.

Copyright © 2017 John Wiley & Sons, Ltd. Dynamic relaxation is an iterative method to solve nonlinear systems of equations, which is frequently used for form finding and analysis of structures that undergo large displacements. It is based on the solution of a fictitious dynamic problem where the vibrations of the structure are traced by means of a time integration scheme until a static equilibrium is reached. Fictitious values are used for the mass and damping parameters. Heuristic rules exist to determine these values in such a way that the time integration procedure converges rapidly without becoming unstable. Central to these heuristic rules is the assumption that the highest convergence rate is achieved when the ratio of the highest and lowest eigenfrequency of the structure is minimal. This short communication shows that all eigenfrequencies become identical when a fictitious mass matrix proportional to the stiffness matrix is used. If, in addition, specific values are used for the fictitious damping parameters and the time integration step, the dynamic relaxation method becomes completely equivalent to the Newton-Raphson method. The Newton-Raphson method can therefore be regarded as a specific form of dynamic relaxation. This insight may help to interpret and improve nonlinear solvers based on dynamic relaxation and/or the Newton-Raphson method. ispartof: INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING vol:113 issue:9 pages:1531-1539 status: published

International audience; Elastic gridshells are structures made of flat two-way grids which are deformed elastically before they are braced and which afterwards mechanically behave like continuous shells. Gridshells present some advantages in terms of manufacturing, lightness and time of assembly. Their covering remains however a technical issue. The present article proposes hence an alternative method to cover them by planar quadrilateral facets, which could also be used as natural bracing if connected properly. It relies on the duality between a certain family of circular meshes with a unique radius and some Tchebycheff nets. The approach is versatile and allows for the design of a large variety of shapes from two curves in space. Real time numerical tools are developed for open and closed curves as well as a strategy for umbilical points. The relaxation of the Tchebycheff net shows finally that an equilibrium configuration can be found in the vicinity of the planar quadrilateral mesh (PQ-Mesh) which confirm the practical feasibility of elastic gridshells covered with planar facets.

Purpose This paper aims to investigate the application of adaptive integration in element-free Galerkin methods for solving problems in structural and solid mechanics to obtain accurate reference solutions. Design/methodology/approach An adaptive quadrature algorithm which allows user control over integration accuracy, previously developed for integrating boundary value problems, is adapted to elasticity problems. The algorithm allows the development of a convergence study procedure that takes into account both integration and discretisation errors. The convergence procedure is demonstrated using an elasticity problem which has an analytical solution and is then applied to accurately solve a soft-tissue extension problem involving large deformations. Findings The developed convergence procedure, based on the presented adaptive integration scheme, allows the computation of accurate reference solutions for challenging problems which do not have an analytical or finite element solution. Originality/value This paper investigates the application of adaptive quadrature to solid mechanics problems in engineering analysis using the element-free Galerkin method to obtain accurate reference solutions. The proposed convergence procedure allows the user to independently examine and control the contribution of integration and discretisation errors to the overall solution error. This allows the computation of reference solutions for very challenging problems which do not have an analytical or even a finite element solution (such as very large deformation problems).

Analysis of three-dimensional frames is a complex process. Each node of these structures has six degrees of freedom, which results in a large set of governing equations. Investigators have utilized the computer power to extend the application of numerical approaches. One of these methods is named dynamic relaxation technique. This strategy explicitly solves the simultaneous system of equations. In this scheme, the static structural equilibrium equations are converted into a dynamic one by adding fictitious mass and damping. To perform nonlinear geometric analysis of 3D frames, 12 classical DR methods are exploited. Previously, the abilities of these approaches in analyzing these structures have not been compared. In each technique, time step, mass, and damping matrices are obtained based on other researchers' works. The structural behavior is assessed with and without considering the shear deformations. For both cases, the load–displacement curves are depicted. In this article, 11 different frames...

In this study, the Maximum Entropy Meshfree (MEM) method is employed for analysing geotechnical problems involving material nonlinearity, assuming small strains. The efficiency of the MEM method is evaluated through several solution schemes for the global governing equations as well as the local constitutive equations. The conventional implicit approach involving the Newton-Raphson method and an explicit adaptive dynamic relaxation technique are employed for solving the governing equations, while local constitutive equations are solved numerically as well as analytically. Two- and three-dimensional numerical experiments are performed to study the efficiency of different configurations of the solution scheme, which leads to some important conclusions about application of the MEM method in geotechnical problems.

The discretization of the boundary value problem for laminated composite shells is based on the finite difference approach using the regular mesh with the constant grid step and the difference operators of the second order of accuracy. The dynamic relaxation method is proposed for the solution of the nonlinear problem. The evolutionary equations of the dynamic relaxation are constructed, and the optimum parameters of the converging linear iterative process are estimated.

The quasi-static explicit finite element method (FEM) and element free Galerkin (EFG) method are applied to trace the post-buckling equilibrium path of thin-walled members in this paper. The factors that primarily control the explicit buckling solutions, such as the computation time, loading function and dynamic relaxation, are investigated and suggested for the buckling analysis of thin-walled members. Three examples of different buckling modes, namely snap-through, overall and local buckling, are studied based on the implicit FEM, quasi-static explicit FEM and EFG method via the commercial software LS-DYNA. The convergence rate and accuracy of the explicit methods are compared with the conventional implicit arc-length method. It is drawn that EFG quasi-static explicit buckling analysis presents the same accurate results as implicit finite element solution, but is without convergence problem and of less-consumption of computing time than FEM.

Using new approach proposed by Dynamic relaxation (DR) method, buckling analysis of moderately thick Functionally graded (FG) cylindrical panels subjected to axial compression is investigated for various boundary conditions. The mechanical properties of FG panel are assumed to vary continuously along the thickness direction by the simple rule of mixture and Mori-Tanaka model. The incremental form of nonlinear formulations are derived based on First-order shear deformation theory (FSDT) and large deflection von Karman equations. The DR method combined with the finite difference discretization technique is employed to solve the incremental form of equilibrium equations. The critical mechanical buckling load is determined based on compressive load-displacement curve by adding the incremental displacements in each load step to the displacements obtained from the previous ones. A detailed parametric study is carried out to investigate the influences of the boundary conditions, rule of mixture, grading index, radius-to-thickness ratio, length-to-radius ratio and panel angle on the mechanical buckling load. The results reveal that with increase of grading index the effect of radius-to-thickness ratio on the buckling load decreases. It is also observed that effect of distribution rules on the buckling load is dependent to the type of boundary conditions.

In the common DR technique, a jump occurs after the appearance of the snap-through or snap-back points. As a result, this approach cannot trace the equilibrium path. To cure this drawback, a load factor is employed. In this paper, a new procedure for calculating this factor is suggested. For this purpose, the work increment of the external loads is minimized with respect to the load factor. This factor is only dependent on the fictitious parameters of the DR strategy. To show the robustness of this formulation, it is used in the geometric nonlinear analysis of various structures. Based on the number of iterations, numbers of convergence points and total duration analysis, the different methods are ranked. The obtained results prove the high efficiency of the suggested scheme. In other words, the authors' technique is accurate and has more rapid convergence rate, in comparison to the similar algorithms.

The main idea of this study is to investigate and compare the efficiency of the dynamic relaxation methods (DRMs) in the form-finding of tensile membrane structures (TMSs). The form-finding process as a main stage in the design and construction of TMSs has been considered for finding an equilibrium configuration subjected to a specific prestress distribution. DRMs as a pseudo-dynamic analysis are an explicit iterative technique for the nonlinear analysis of structures. In the techniques, the static equilibrium of structures is solved by integrating the pseudo-dynamic equations in order to obtain the steady state of the pseudo-dynamic problem. In this study, the efficiency and generality performance of DRMs are compared through solving three selected TMSs. In order to achieve this purpose, seven schemes of the DR approach are selected based on combining the fictitious parameters including the time step, diagonal mass, and damping matrices which were proposed in the previous studies. Furthermore, a reference index is proposed by combining the total number of iterations and the overall duration of analysis in order to appropriately compare the schemes of the DR approach in the form-finding of TMSs.

The Analytic Target Cascading (ATC) is an effective method for solving hierarchical Multidisciplinary Design Optimization (MDO) problems. At the same time, this method suffers from poor convergence and low accuracy, which is caused by the inconsistency of system constraints. In this paper, a novel ATC method based on dynamic relaxation factor is proposed. The dynamic relaxation factor of consistency constraint is added in the system level and is adjusted by the deviation of the linking variables between the levels to ensure the feasible region of the design space. The effectiveness and accuracy of this method are verified by a mathematical example. This method is used to solve the lightweight problem of the trussed front part of the vehicle body frame based on active and passive safety to achieve the collaborative optimization of lightweight trussed frame, crash safety, and aerodynamic characteristics. The important value of the novel ATC method based on dynamic relaxation factor in engineering applications is proven.

The buckling and large deflection behaviors of axis-symmetric radially functionally graded (RFG) ring-stiffened circular plates are investigated by the dynamic relaxation (DR) method combined with the finite difference discretization technique. The material properties of the constituent components of the RFG plate are assumed to vary continuously according to the Mori-Tanaka distribution along the radial direction. The nonlinear governing equations are obtained in the incremental form based on the first-order shear deformation plate theory (FSDT) and the von Karman relations for large deflection. In the buckling analysis, an external in-plane load is applied to the plate incrementally so that, in each load-step, the incremental form of the governing equations can be solved by a numerical code prepared based on the DR method. After converging the DR code in the first increment, the latter load-step is added to the previous one, and the program is repeated again. The critical buckling load is determined from the compressive load-displacement curve obtained by solving the incremental form of the governing equa- tions. Based on the present incremental form of formulation, a bending analysis can also be conducted if the whole load is applied simultaneously. Finally, a detailed parametric study is carried out to investigate the influences of various boundary conditions, grading indices, thickness-to-radius ratios, stiffener’s positions and depths on the critical buckling load, and displacements and stresses resulted from the bending analysis. It is observed that the effect of the stiffener on the results is much greater in the functionally graded plate with higher material grading indices. The results also reveal that, by increasing the depth of the stiffer, the values of ascending the critical buckling load are approximately identical for both simply supported and clamped boundary conditions.

The numerical structural analysis schemes are extensively developed by progress of modern computer processing power. One of these approximate approaches is called "dynamic relaxation (DR) method." This technique explicitly solves the simultaneous system of equations. For analyzing the static structures, the DR strategy transfers the governing equations to the dynamic space. By adding the fictitious damping and mass to the static equilibrium equations, the corresponding artificial dynamic system is achieved. The static equilibrium path is required in order to investigate the structural stability behavior. This path shows the relationship between the loads and the displacements. In this way, the critical points and buckling loads of the non-linear structures can be obtained. The corresponding load to the first limit point is known as buckling limit load. For estimating the buckling load, the variable load factor is used in the DR process. A new procedure for finding the load factor is presented by imposing the work increment of the external forces to zero. The proposed formula only requires the fictitious parameters of the DR scheme. To prove the efficiency and robustness of the suggested algorithm, various geometric non-linear analyses are performed. The obtained results demonstrate that the new method can successfully estimate the buckling limit load of structures.

A new variable load factor is proposed for Dynamic Relaxation Method.This formula is based on the minimizing of the unbalanced displacement.To prove the capability and efficiency of the presented scheme, several numerical tests are performed.The results indicate that the suggested approach can trace the complex structural static paths. Dynamic Relaxation (DR) method is an explicit iterative technique suitable for nonlinear structural analysis. In this method, the static equilibrium equations are converted to a fictitious dynamic system. In general, DR iterations are unstable. Nevertheless, the fictitious parameters, such as the diagonal mass and damping matrices as well as the time steps are determined so that the stability conditions are satisfied. If the mass and damping matrices are selected properly, the structural responses converge to the accurate static solutions. Nonlinear behavior of structures includes various characteristics, such as the existence of load limit points; displacement limit points, buckling points, post-buckling region and bifurcation points. All of these valuable features are demonstrated by the equilibrium path, and most of the DR procedures cannot completely trace it. To overcome this weakness, a new variable load factor is presented by minimizing the unbalanced displacement. In the second suggested algorithm, the load factor is determined based on the parts of the structural equilibrium path placed between two limit points. In order to prove the capability of the proposed strategies, several 3D trusses and 2D frames, with geometrical nonlinear behavior, are analyzed. The numerical results indicate that the proposed approach can trace the complex structural equilibrium path.

A method for form finding with dynamic relaxation and Non-Uniform Rational B-Splines (NURBS) based isogeometric membrane elements has been implemented and studied regarding the influence of the discretization and element shape on the form finding. The procedure allows for rapid form finding since NURBS describe the curved geometry well and it is shown that the form-finding can be performed using a coarse mesh. However, to minimize the bending strain energy a fine mesh is needed. Using smaller elements is more advantageous than increasing the degree of the basis functions, keeping the number of integration points few and converging at a lesser number of iterations. Using isogeometric analysis (IGA) simplifies further studies since the form-found shape can be exactly represented with a shell element formulation. The method is suitable to be used in computer aided design environments such as Rhinoceros 3D during design stages, where the form finding can be evaluated together with other studies in a design context.

The problem of finding the un-deformed configuration of an elastic body, when the deformed configuration and the loads are known, occurs in many engineering applications. Standard solution methods for such problems include conservation laws based on Eshelby’s energy–momentum tensor and re-parameterization of the standard equilibrium equations. In this paper we present a different method for solving such problems, based on a re-parameterization of the nodal forces using the Total Lagrangian formulation. The obtained nonlinear system of equations describing equilibrium can be solved using either Newton–Raphson or an explicit dynamic relaxation algorithm. The solution method requires only minor modifications to similar algorithms designed for forward motion calculations. Several examples involving large deformations and different boundary conditions and loads are presented.

In this paper the nonlinear bending of laminated stiffened annular sector plates under mechanical loading with various boundary conditions is investigated. The plates are made of layers with orthotropic properties and different fiber orientations, which the aforementioned fibers are placed in a Cartesian coordinate system. Based on first-order shear deformation plate theory (FSDT) and von Karman relations for large deflection, nonlinear equilibrium equations are developed. Dynamic relaxation (DR) numerical method combined with the finite difference discretization technique is used to solve the plate nonlinear partial differential equations and FORTRAN program is developed to generate the numerical results. Effects of the plate thickness-to-radius ratio, boundary condition, stiffener depth, plate lay-ups and the sector angle are discussed.

This paper presents a method for the optimization of multi-component structures comprised of two and three materials considering large motion sliding contact and separation along interfaces. The structural geometry is defined by an explicit level set method, which allows for both shape and topology changes. The mechanical model assumes finite strains, a nonlinear elastic material behavior, and a quasi-static response. Identification of overlapping surface position is handled by a coupled parametric representation of contact surfaces. A stabilized Lagrange method and an active set strategy are used to model frictionless contact and separation. The mechanical model is discretized by the extended finite element method which maintains a clear definition of geometry. Face-oriented ghost penalization and dynamic relaxation are implemented to improve the stability of the physical response prediction. A nonlinear programming scheme is used to solve the optimization problem, which is regularized by introducing a perimeter penalty into the objective function. Sensitivities are determined by the adjoint method. The main characteristics of the proposed method are studied by numerical examples in two dimensions. The numerical results demonstrate improved design performance when compared to models optimized with a small strain assumption. Additionally, examples with load path dependent objectives display non-intuitive designs.

[EN] In the field of bending-active structures, the complexity of finding beforehand the equilibrium configuration and the non-linearity of the structural response are main issues during the conceptual phase. The use of tools based on classical form-finding procedures as dynamic relaxation is the main trend today; different mechanical models with 3, 4 or 6 degrees of freedom have been implemented for modelling the bending effect. However, there is a well-established class of mechanical models which has been specifically designed to reproduce the behaviour of very flexible structures and has not been used so far in form-finding of bending-active structures. These are derived from the so-called geometrically exact (or Reissner-Simo) beam theory, and they are able to treat arbitrarily large rotations and displacements. In this paper, we present the development of a form-finding tool based on Reissner-Simo¿s theory and the dynamic relaxation method, in order to find the static equilibrium of the system. The choice of form-finding parameters as the target curve length and the kinematic constraints at beam ends will determine the shape of the final structure in the `design-oriented¿ process. Several numerical examples on a range of structures are tested to validate the formulation., The financial support from the Spanish Ministry of Economy and Competitiveness through grant BIA2015-69330-P (MINECO) is gratefully acknowledged.

This paper proposes an extended design concept and mechanical description for cable-net structures, including 10 states and 15 procedures which are defined according to their physical nature and analytical capabilities. In the pre-stress release analysis, an iterative computational method is developed for the inverse evaluation from the equilibrium state to the zero-stress state, which adopts the least norm least square approach (LNLS) to the compatibility equation because of the indeterminate property of a cable-net structure. In the pre-tensioning development analysis, another iterative computational method is developed for the positive problem from the zero-stress state to the actual pre-stress state by moving the boundary joints, in which the explicit governing equations are formulated based on the particular energy function and a feasible self-stress mode is adopted to avoid the singularity of the initial stiffness matrix. To implement these methods, Matlab algorithms are developed and two examples are investigated. By comparing the results of the iterative method with those of the dynamic relaxation method, this study determines that they are comparable with each other, which validates the efficiency and accuracy of these iterative methods.

In this paper we introduce a new multi-vector update quasi-Newton (MVQN) method for implicit coupling of partitioned, transient FSI solvers. The new quasi-Newton method facilitates the use of ‘black-box’ field solvers and under certain circumstances can be demonstrated to provide Newton-like convergence behaviour for strongly coupled FSI benchmark problems. We demonstrate the superior convergence behaviour and robust nature of the MVQN method compared to other well known quasi-Newton coupling schemes, including the least squares reduced order modelling (IBQN-LS) scheme, the classical rank-1 update Broyden’s method and fixed point iterations with dynamic relaxation. The quasi-Newton methods are analysed on a suite of strongly coupled FSI problems, including but not limited to, internal, incompressible flow through a flexible tube where the solid density is an order of magnitude lower than the fluid density.

Dynamic relaxation (DR) is an iterative method for solving simultaneous sets of equations, which govern the behavior of static structures. One of the most effective techniques in this category is the kinetic damping approach. In this research, a new formula for the fictitious time step is obtained by error analysis to utilize in the kinetic DR method. The developed algorithm is applied to the nonlinear analysis of several plane- and space-trusses, and also structural frames. Numerical results indicate that the proposed fictitious time step has a significant effect on improving the convergence rate of the kinetic DR method.

Hybrid arc-length methods have been used for tracing the post-buckling equilibrium path of semirigid elastoplastic space frames. For example, the original implicit arc-length method uses the implicit Newton–Raphson method in both the predictor and the corrector steps, while the explicit arc-length method uses the explicit dynamic relaxation method in both the steps. The explicit and implicit arc-length methods have a clear disadvantage in that both require an excessive number of iterations, and the matrices are often singular. In this study, algorithms for implicit–explicit and explicit–implicit hybrid arc-length methods are developed for use in the predictor and corrector steps, to improve the accuracy and efficiency of the said method. The accuracy and applicability of the proposed methods are investigated by solving examples.

A new method to assess the raking performance of Platform timber framed walls, is provided in this study: each component of the unit wall assembly is assumed as rigid, hence allowing to drastically reduce the overall number of DoFs involved within the model. The timber frame in particular, is modelled as a mechanism, having only two DoFs (regardless of the number of studs) corresponding to the horizontal and rotational displacements of the header beam. For a given imposed horizontal displacement Δ h , the corresponding racking load P ( Δ h ) is computed by numerical relaxation, allowing to consider a continuous function to represent the load-slip curves of the connections. A comparison of the numerical analysis against laboratory test results is provided, showing the method’s capability in predicting the raking strength of the wall, despite the assumed reduced number of DoFs.

Phase changes and heat exchanges are examples of physical processes appearing in many industrial applications involving multiphase compressible flows. Their knowledge is of fundamental importance to reproduce correctly the resulting effects in simulation tools. A fine description of the flow topology is thus required to obtain the interfacial area between phases. This one is responsible for the dynamics and the kinetics of heat and mass transfer when evaporation or condensation occurs. Unfortunately this exchange area cannot be obtained easily and accurately especially when complex mixtures (drops, bubbles, pockets of very different sizes) appear inside the transient medium. The natural way to solve this specific trouble consists in using a thin grid to capture interfaces at all spatial scales. But this possibility needs huge computing resources and can be hardly used when considering physical systems of large dimensions. A realistic method is to consider instantaneous exchanges between phases by the way of additional source terms in a full non-equilibrium multiphase flow model [2,15,17]. In this one each phase obeys its own equation of state and has its own set of equations and variables (pressure, temperature, velocity, energy, entropy,...). When enabling the relaxation source terms the multiphase mixture instantaneously tends towards a mechanical or thermodynamic equilibrium state at each point of the flow. This strategy allows to mark the boundaries of the real flow behavior and to magnify the dominant physical effects (heat exchanges, evaporation, drag,...) inside the medium. A description of the various relaxation processes is given in the paper. Les changements de phase et les transferts de chaleur sont des exemples de phénomènes physiques présents dans de nombreuses applications industrielles faisant intervenir des écoulements compressibles multiphasiques. La connaissance des mécanismes associés est primordiale afin de reproduire correctement leurs effets à travers des outils de simulation. Dans ce cadre, une description fine de la topologie d’un écoulement est nécessaire afin de connaître précisément l’aire interfaciale entre toutes les phases. Celle-ci est en effet responsable de la dynamique et de la cinétique des transferts de masse et de chaleur lorsque de l’évaporation et de la condensation se produisent. Malheureusement cette aire interfaciale est difficilement accessible particulièrement lorsque des mélanges complexes se forment (gouttes, bulles, inclusions de différentes tailles) au sein du milieu. La façon la plus naturelle de résoudre ce problème est d’utiliser un maillage suffisamment fin afin de capturer toutes les interfaces présentes à toutes les échelles. Cependant cette possibilité demanderait des ressources informatiques démesurées au vue de certains systèmes pouvant être de très grande taille. Une méthode plus réaliste est de considérer que les échanges entre les phases s’effectuent instantanément. Des termes sources de relaxation liés à ces échanges sont utilisés dans un modèle d’écoulement compressible à phases séparées en déséquilibre [2,15,17]. Dans celui-ci, chaque phase possède son propre jeu d’équations et ses propres variables (pression, vitesse, température, énergie, entropie, ...). Quand les termes de relaxation sont activés, le mélange multiphasique tend instantanément en chaque point de l’écoulement vers un état d’équilibre prédéfini. Cette approche permet également de borner les conditions réelles d’écoulement et de souligner les effets physiques dominants (transfert de chaleur, évaporation, trainée, ...). Une description des différents processus de relaxation est proposée dans ce papier.

In this study the large deflection behaviors of stiffened annular functionally graded (FG) sector plates under mechanical and thermo-mechanical loadings with various boundary conditions are investigated. Material properties are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Based on first-order shear deformation plate theory (FSDT) and von Karman relations for large deflection, nonlinear equilibrium equations are developed. Dynamic relaxation (DR) numerical method combined with the finite difference discretization technique is used to solve the plate nonlinear equations. Effects of material grading index, boundary condition, stiffener depth to the plate thickness ratio and thermal gradient are discussed.

This paper proposes and validates a three degrees of freedom element formulation that accounts for torsion and transverse bending of three-dimensional curved elements in explicit numerical analyses methods such as Dynamic Relaxation. Using finite difference modeling, in-plane distortions and moments, the increment of twist (and hence torsion) and out-of-plane bending deformations are determined. Numerical stability and convergence limits of the element are discussed. Two sets of circular arc test cases validate the accuracy of the element against theoretical and six degrees of freedom finite-element results. This element is widely applicable and found in strained grid shells and spline stressed membranes.

Twelve new and well known dynamic relaxation methods are considered. Several benchmark frame and truss structures, with geometric nonlinear behavior, are solved by these schemes. Based on the total number of iterations and overall analysis duration, the studied techniques are graded. In this way, the efficiencies of the solvers are found and these algorithms are sorted in terms of their abilities.

We put forward a hydrodynamic theory of nematic liquid crystals that includes both anisotropic elasticity and dynamic relaxation. Liquid remodeling is encompassed through a continuous update of the shear-stress free configuration. The low-frequency limit of the dynamical theory reproduces the classical Ericksen-Leslie theory, but it predicts two independent identities between the six Leslie viscosity coefficients. One replicates Parodi's relation, while the other-which involves five Leslie viscosities in a nonlinear way-is new. We discuss its significance, and we test its validity against evidence from physical experiments, independent theoretical predictions, and molecular-dynamics simulations., 6 pages, 1 figure, 2 tables

Solving nonlinear equations has become increasingly significant and indispensable in many disciplines. In the recent decade, two powerful methods, the Zhang dynamics (ZD) method and the gradient dynamics (GD) method have been extensively investigated and implemented in solving nonlinear equations. In addition, some conventional methods, such as the dynamic relaxation method (DRM), redundant manipulators dynamics (RMD), and Langevin dynamics (LD), can also be applied to solve nonlinear equations. In this paper, we propose four novel types of systems based on the ZD method, the GD method and the hybrid dynamics methods to solve nonlinear equations in time-invariant and time-variant situations, respectively. In addition, it is worth pointing out that, by comparison, the four proposed systems depicted in the second-order dynamics are evidently quite different from and more general than some other conventional systems/dynamics for solving nonlinear equations, which all constitute the unified second-order methodology for nonlinear equations solving.

The application of tensegrity structures has been extended to deployable structures, smart structures, and robots. In these applications, large displacement shape control is usually involved. The problem of finding a collision-free path for the shape control emerges. In this article, a preliminary study on the collision-free shape control of tensegrity structures is carried out based on a representative two-dimensional tensegrity system. An incremental procedure based on dynamic relaxation method is proposed to track the trajectory of the movement during shape control. For every increment along the trajectory, collision is checked by a method based on geometrical analysis. The effect of the size of the obstacle to the number of feasible paths is investigated. It is found that when the size of the obstacle reaches a critical value, the structure is no longer able to move to the target configuration by a single step. As a result, a relay point is added in the free space. Via the relay point, free paths connecting the initial configuration and the target configuration are found.

This paper focuses on the dynamic relaxation (DR) method as an efficient approach for solving a system of simultaneous equations. This is an iterative procedure which can be used for both finite element and finite difference structural analysis. The DR method has a simple algorithm. However, it suffers from low convergence rate. In the current study, a residual energy minimizer timestep (REMT) will be formulated by minimizing the residual energy. A variety of structural analyses with linear and nonlinear (elastic large deflection) behaviors demonstrate the potential of the proposed strategy. The results indicate that the REMT improves the convergence rate of DR without any additional constraints so that the cost and computational time are decreased.

Highlights? A new fictitious mass derives for kinetic DR method based on incremental analysis. ? This formulation combines with the transformed Gerschgorin theory presented here. ? The proposed scheme is used for static and dynamic analyses of structures. ? The convergence rate of the new method is higher than the other DR approaches. This paper deals with the stability and convergence rate of the kinetic damping process in Dynamic Relaxation (DR) method. Based on running an incremental analysis, a new relationship is presented for fictitious mass of the kinetic DR algorithm. This formulation is also combined with a transformational form of the Gerschgorin circles theory, proposed here. For numerical verification, some structures from the finite element models are analyzed statically and dynamically and results are compared with the well-known viscous and kinetic DR algorithms. These examples clearly show that the proposed fictitious mass considerably increases the convergence rate of the kinetic DR iterations.

The explicit nonlinear dynamic relaxation method (DRM) is applied to the nonlinear geodesic shape finding analysis by introducing fictional tensioned \'strings\' along the desired seams with a three or four-node membrane element. A number of results from the numerical example for the nonlinear geodesic shape finding and patterning analysis are obtained by the proposed method to demonstrate the accuracy and efficiency of the developed method. Therefore, the proposed geodesic shape finding algorithm may improve the applicability of a four-node membrane element to membrane structural engineering and design analysis simultaneously for the shape finding, stress, and patterning analysis.

The main idea of the paper is to present a dynamic relaxation algorithm that does not require the damping matrix and velocity terms. The general formulation suggested in this article covers the common DRM as well. In order to verify the ability of the new technique, the obtained static and dynamic solutions are checked with the ones found by the other scheme. When the loads are variable and the behavior of the system is extremely nonlinear, the proposed procedure works efficiently.

Proper shape analysis of spatial cable-frames for purpose of preliminary design is discussed in this paper. By functional analysis or using the equilibrium equation group directly, curve functions of the upper and lower chord cables can be built. When only the self-equilibrated internal force is considered in the system without external loads and self-weights of members, optimal curve shape of chord cables are straight lines. The numerical algorithm by bonding linear deduction method and dynamic relaxation method for wind load adaptive shape analysis of spatial cable frames is put forward. The adaptive shape determination of spatial cable frames with averaged wind load of a practical stadium canopy is tried.

This paper develops a new method for calculating the viscous fictitious damping of the dynamic relaxation (DR) method to overcome one of the most crucial difficulties in its application - the low convergence rate. The DR formulation was derived by error minimizations between two successive iterations to deduce an optimum fictitious mass and viscous damping with the aid of the Stodola iterative process. The efficiency of the new method was verified by its application to a wide range of typical structures with strong nonlinearity. The results show that compared to the conventional DR algorithm such as kinetic approach, the new method improves the convergence rate considerably.

Combining geometrical characteristics analysis and mechanical properties optimization, this paper presents an optimization approach for free-form space structures based on natural soap film property. The quasi-uniform B-spline surface is applied to express the initial surface, and the prediction technology of tangent vectors is adopted to satisfy the specific geometric configuration requirements. To establish the rational objective surface, the initial surface is treated as soap film and optimized by Young–Laplace equation and dynamic relaxation method. Numerical experimental results demonstrate that the proposed optimization approach is convenient and feasible, and it is effective for shortening the design period of free-form space structures.