A tutorial on simulating nonlinear behaviors of flexible structures with the discrete differential geometry (DDG) method

On discrete differential geometry in twistor space

This work is concerned with the study of thin structures in Computational Mechanics. This field is particularly interesting, since together with traditional finite elements methods (FEM), the last years have seen the development of a new approach, called discrete differential geometry (DDG). The idea of FEM is to approximate smooth solutions using polynomials, providing error estimates that establish convergence in the limit of mesh refinement. The natural language of this field has been found in the formalism of functional analysis. On the contrary, DDG considers discrete entities, e.g., the mesh, as the only physical system to be studied and discrete theories are being formulated from first principles. In particular, DDG is concerned with the preservation of smooth properties that break down in the discrete setting with FEM. While the core of traditional FEM is based on function interpolation, usually in Hilbert spaces, discrete theories have an intrinsic physical interpretation, independently from the smooth solutions they converge to. This approach is related to flexible multibody dynamics and finite volumes. In this work, we focus on the phenomenon of membrane locking, which produces a severe artificial rigidity in discrete thin structures. In the case of FEM, locking arises from a poor choice of finite subspaces where to look for solutions, while in the DDG case, it arises from arbitrary definitions of discrete geometric quantities. In particular, we underline that a given mesh, or a given finite subspace, are not the physical system of interest, but a representation of it, out of infinitely many. In this work, we use this observation and combine tools from FEM and DDG, in order to build a novel discrete shell theory, free of membrane locking.

Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. This survey contains a discussion of the following two fundamental discretization principles: the transformation group principle (smooth geometric objects and their discretizations are invariant with respect to the same transformation group) and the consistency principle (discretizations of smooth parametrized geometries can be extended to multidimensional consistent nets). The main concrete geometric problem treated here is discretization of curvature-line parametrized surfaces in Lie geometry. Systematic use of the discretization principles leads to a discretization of curvature-line parametrization which unifies circular and conical nets.

Static analysis of elastic cable structures under mechanical load using discrete catenary theory

Thermal protection system (TPS) composed of ceramic insulation tile and strain-isolation pad (SIP) tends to be destroyed by excessive inner strain arises from aerodynamic load or substructure deformation/vibration. Both material nonlinearity and geometric nonlinearity affect the static and dynamic strength of TPS, and the nonlinear effects of SIP need to be considered in the strength analysis. By considering both the material nonlinearity and geometric nonlinearity of SIP, the effect of external static load on the dynamic characteristics of tile-SIP system is investigated. Firstly, a theoretical model is established to calculate the natural frequency of tiles under static load with material and geometric nonlinearities. The analysis procedure is separated into two steps. The first step is the nonlinear analysis to determine the static large deformation. The second step is the linearized dynamic analysis on the base of static deformation with small vibration amplitude assumption. The complexity is reduced by introducing a nonlinear coefficient parameter. The theoretical results have good agreement with COMSOL simulation. Numerical results show that large deformation is found in the flexible SIP when uniform static load applied on the surface of the tile, which introduces significant material and geometric nonlinearities and changes the dynamic characteristics of the tile. 9% increase of intrinsic frequency is found for the static load of 80kPa when considering material nonlinearity, while 16% increase of intrinsic frequency is found when considering both material and geometric nonlinearities. It is concluded that the uniform static load has important effect on the nature frequency of tile-SIP systems when introducing the material and geometric nonlinearities into the calculation. The nonlinearities of the SIP need to be considered in the tile-SIP dynamic strength analysis.

Contact dynamic analysis of tether-net system for space debris capture using incremental potential formulation

Exact prediction of the mechanical behavior of nano-sensors and nano-actuators directly depends on the models applied to analyze their nano-components. From their dynamic behavior point of view, despite many studies related to the geometrical nonlinearities in modeling the nanostructures, one of the main issues that has not been addressed appropriately is the effects of material nonlinearity. Hence, this paper intends to fill this gap and deals with an investigation of combined geometrical and material nonlinearities on the nonlinear dynamic response of the embedded nanobeams in their free vibration as well as the primary and superharmonic resonances. The material nonlinearity considered in this research is formulated based on the sum of linear and cubic relations between the stress and strain. The results reveal that, for the material and boundary conditions of the embedded nanobeam studied in this research, the material nonlinearity tries to cause softening effects while the geometrical nonlinearity attempts to make hardening influences. The competition between these two effects leads to interesting nonlinear behaviors of the nanobeam for different dimensions and dynamic regimes that are scrutinized in this paper. In addition, the effects of applying external magnetic fields on the nonlinear responses of the nanobeams are investigated.

In the article the main aspects of the formation of flexible management structures of corporate associations of construction enterprises — construction groups are considered. The subjects of the investment and construction complex must quickly respond to changes in the external environment, should have possibility to adjust to possible changes in the economy and other spheres. Such an opportunity can be realized on the basis of the formation of flexible management structures of organizations. A flexible management structure is an organizational structure that allows an enterprise to respond quickly and effectively to the changes in the external or internal environment. In general, flexible management structures are a set of system elements that can adapt to changing environmental conditions in order to ensure an acceptable level of efficiency and production optimization and management processes under conditions of changing environment. In this case, the formation of organizational structures is implemented at the following levels: intra-company level, organizational and functional level, organizational and contractual level. Formation of flexible organizational structures for the management of a construction group should be realized within the limits of the certain strategy, while fulfilling the conditions for maximizing the level of strategic development while ensuring a sufficient level of general stability of the construction group.

As aircraft wings become more flexible as a result of searching for more fuel efficient and higher performance solutions, structural nonlinearities become more apparent. Geometric nonlinearities make the structure’s modal parameters a function of the deformed shape and therefore of the loading condition. Modal characterization of very flexible structures is challenging due to these nonlinearities and the very low natural frequencies (the fundamental mode is typically below 1 Hz). In traditional, stiffer structures, a single shape is sufficient to characterize the structure through ground vibration testing and finite model updating due to its linear behavior. However, with very flexible structures, different deformed shapes have different modal parameters (frequencies, damping, and mode shapes). This paper investigates the impact of large displacements on the modal parameters (frequency and mode shapes) of very flexible structures and introduces a method to update the corresponding finite element model. Results are presented to discuss the impact of deformed shapes and the development and applicability of the procedure to general very flexible wing structures.

Decohesion undergoing large displacements takes place in a wide range of applications. In these problems, interface element formulations for large displacements should be used to accurately deal with coupled material and geometrical nonlinearities. The present work proposes a consistent derivation of a new interface element for large deformation analyses. The resulting compact derivation leads to a operational formulation that enables the accommodation of any order of kinematic interpolation and constitutive behavior of the interface. The derived interface element has been implemented into the finite element codes FEAP and ABAQUS by means of user-defined routines. The interplay between geometrical and material nonlinearities is investigated by considering two different constitutive models for the interface (tension cut-off and polynomial cohesive zone models) and small or finite deformation for the continuum. Numerical examples are proposed to assess the mesh independency of the new interface element and to demonstrate the robustness of the formulation. A comparison with experimental results for peeling confirms the predictive capabilities of the formulation.

Nonlinearity in hall devices and its compensation

An elasto-plastic finite element procedure using degenerated shell element with assumed strain field technique considering both material and geometric nonlinearities has been developed. This assumes von-Mises yield criterion, von-Karman strain displacement relations and isotropic hardening. A few numerical examples are presented to demonstrate the correctness and applicability of the method to different kinds of engineering problems. From present study, it is seen that there is a considerable improvement in the displacement valuse when both material and geometric nonlinearities are considered. An example of the spread of plastic zones for isotropic and anisotropic materials has been illustrated.

State-of-Art review on deployable scissor structure in construction

This study proposes a time-domain spectral finite element (SFE) method for simulating the second harmonic generation (SHG) of nonlinear guided wave due to material, geometric and contact nonlinearities in beams. The time-domain SFE method is developed based on the Mindlin–Hermann rod and Timoshenko beam theory. The material and geometric nonlinearities are modeled by adapting the constitutive relation between stress and strain using a second-order approximation. The contact nonlinearity induced by breathing crack is simulated by bilinear crack mechanism. The material and geometric nonlinearities of the SFE model are validated analytically and the contact nonlinearity is verified numerically using three-dimensional (3D) finite element (FE) simulation. There is good agreement between the analytical, numerical and SFE results, demonstrating the accuracy of the proposed method. Numerical case studies are conducted to investigate the influence of number of cycles and amplitude of the excitation signal on the SHG and its performance in damage detection. The results show that the amplitude of the SHG increases with the numbers of cycles and amplitude of the excitation signal. The amplitudes of the SHG due to material and geometric nonlinearities are also compared with the contact nonlinearity when a breathing crack exists in the beam. It shows that the material and geometric nonlinearities have much less contribution to the SHG than the contact nonlinearity. In addition, the SHG can accurately determine the crack location without using the reference data. Overall, the findings of this study help further advance the use of SHG for damage detection.

In this study, the nonlinear free and forced vibrations of a nanobeam resting on a viscoelastic medium have been investigated. The quadratic material nonlinearity, which is neglected in the literature, is included in the stress–strain relation, and then using nonlocal elasticity theory, the nonlinear vibration of the nanobeam considering both material and geometrical nonlinearities is investigated. To this end, the governing equation of motion is extracted using the Euler–Bernoulli beam theory and utilizing Hamilton’s principle. Applying Galerkin’s method, the nonlinear differential equation of the nanobeam is obtained. The cubic and quantic nonlinearities in the governing differential equation are due to geometrical and material nonlinearities, respectively. Using Modified Homotopy Perturbation Method, the nonlinear differential equation is solved, and the nanobeam time response and frequency are obtained. Also, the frequency response of nanobeam in the presence of a viscoelastic medium and harmonic excitation is obtained. The results illustrate that material nonlinearity has an important effect on the free and forced vibration responses of the system. For validation, the obtained results of nanobeam are compared with the results obtained from the fourth-order of Runge Kutta numerical method and previous research.