Comparison of several dynamic relaxation methods in the case of large displacements of membrane structures

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Kinetic damping

Before presenting the dynamic relaxation method itself, we present the kinetic damping. Kinetic damping method was firstly introduced by Cundall [32]. It consists in resetting velocity to zero at each kinetic energy peak. This simple procedure has generally been found to be stable and allowing a fast convergence [30]. When there is no external energy acting, there is just internal elastic energy to evolve towards the steady state position. Velocities are reset to zero at each kinetic energy peak because the structure oscillates, and the kinetic energy peaks are supposed to occur close to the static equilibrium position. Then, the computation is restarted from the current configuration, but with zero initial velocity. This procedure is continued until the structure comes progressively to a static equilibrium position.
An example of application that will be presented and developed later on can be seen in the Figure 3.1, where it can be observed in red the evolution of the kinetic energy and in green the residual. There, we can see that the residual decreases significantly each time the velocities are reset to zero, what means that the method is effective. However, it can also be observed that the effectiveness is inferior at the end of the calculation, where kinetic energy peaks are very small. Figure 3.1: Kinetic energy and residual for a random DR calculation where kinetic damp-ing has been applied A detailed review of the implementation of kinetic damping can be read in the works of Barnes [30, 36] or even in the thesis of Troufflard [1]. This damping method was implemented in the calculation software used in this thesis (Herezh++) by including several parameters, that are detailed in the following: Kinetic damping parameters implemented and used in Herezh++ Although the kinetic damping principle is very simple, the numerical behavior of the calculations makes it interesting to have the control of some parameters concerning the kinetic damping. In this section, the input parameters used in Herezh++ are described. (NOTE: Unless if a detailed description of the used parameters is wanted, this part can be skipped).
Kinetic damping does not have to be applied from the very beginning of the calculation. To avoid it to start right at the beginning and thus avoiding taking into account eventual initial oscillations, a parameter can be defined to indicate since which iteration we need it to start working:
nb_deb_test_amor_cinetique_ [iteration_number].
During a calculation, a high number of oscillations can appear. In order to avoid the kinetic damping to act in all the ”fake” kinetic energy peaks, there is the possibility of defining a parameter that indicates how many times the kinetic energy has to decrease in order to allow to reset velocities to 0: max_nb_decroit_pourRelaxDyn_ [number].
For the same reason of the eventual many oscillations, it might result advantageous not to look at the kinetic energy at every single iteration, but calculating the average over a bunch of iterations, and resetting to zero the velocities whenever a peak is reached in this averaged value of kinetic energy. A parameter allows to indicate the number of iterations considered to calculate the average (by default, if the parameter is omitted, its value is 1): taille_moyenne_glissante_ [number_of_iterations].
It is difficult to determine with precision the peaks of kinetic energy, and a criterion can be used to ensure a peak has been reached. For example, we can assume that the kinetic energy has passed a peak whenever it diminishes after a peak a certain value. This value can be a fraction of the last peak. In spite of that, a parameter is defined to indicate the value of the fraction, that can be adjusted according to our interest: test_fraction_energie_ [fraction_value].

Dynamic relaxation method

The problem that we need to solve, after discretization by FE, is: [M]X + R(X, X) = 0. where [M ] is the diagonal mass matrix; X, X , X are respectively the position, velocity ˙ and acceleration of the nodes and R(X, X) is the residual of internal and external forces in function of the position and velocity of nodes. The term [M ]X represents the generalized expression of the acceleration forces. The method used to solve the problem in time is based on the explicit centered finite differences method (CFD). The calculation of equilibrium is always made in the final configuration, using the Cauchy’s stress tensor. In the case of a Hooke’s elastic law of behavior, Cauchy’s tensor is associated with Almansi’s deformation measure. The deformation is calculated from the variation of the 1The # symbol means ”commented line”. Therefore, the two commented lines mean that we did not use those two parameters in our calculations coordinates of the metric tensor associated to the material coordinates of the point in the reference element (a comprehensive example of this calculus has been presented in [20]). Then, the stress tensor is obtained by means of the formulae:
Spheric part : trace(σ) = K trace(ǫ) (3.2) Deviatoric part : dev(σ) = 2 G dev(ǫ) (3.3) In the case of the more complex material behavior: Hart-Smith, that we use in one of our numerical examples, the tensor of left Cauchy-Green B is determined, using again the coordinates of the metric tensor. Its invariants are then used to define the potential and its different variations, needed for the calculation of Cauchy’s stress tensor [20]. For both of these behaviors, only the initial and final states are taken into account.
Since acceleration forces depend on the mass, the dynamic relaxation method proposed by Barnes [36] uses an arbitrary mass term in order to improve the kinetic damping while keeping the numerical stability. This mass matrix is obtained from the Courant-Friedrichs-Lewy (CFL) condition [8] which gives the maximum Δt. This limitation is given by: Δt ≤ 2 (3.4) ωmax where ωmax is the maximum frequency of equation (3.1). This frequency can be expressed by: ωmax = √ (3.5).

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Incremental scheme and convergence criterion

In the case of an incremental law of behavior, a priori not totally reversible, when the loading leads to big deformation-stress final states, the final-form finding procedure in one step is not correct anymore. The final state depends indeed on the loading path which in the case of DR can be very different to the real path. A solution is to use an incremen-tal loading procedure. Assuming that increments are small enough, the procedure then guarantees a succession of points of static physical equilibrium that allows to be close to the real response of the structure during the loading. The convergence criterion in the steady state must comply with two points. In one part, the structure must be in mechanical equilibrium, what we represent as a norm of the residual of the static generalized forces -internal and external-, inferior to an instruction’s value. In other part, in the case where the kinematic boundary conditions block the global solid movements, we impose the final kinetic energy to be also inferior to an instruction value, what also guarantees the equilibrium in the case where the generated stresses are very low. In the practical, these conditions are applied under a relative form according to.

Numerical case studies

In this section, we will perform numerical case studies on the formulae 3.12 and 3.14. We use the C++ academic finite elements software Herezh++ [18], and for the meshing and postprocessing, we use the software Gmsh [66]. The numerical case studies in the section are described below:
• Firstly, we show how we can adapt the parameters of the first of our proposed formulae to obtain an equivalent calculation to Han and Lee’s formulation. We also compare it to our second formule using the classical test of inflation of a rectangular cushion.
• Based on the same numerical test, we show that both of our proposed formulae work with complex meshes, with different geometries, with different element types and different interpolations.
• We demonstrate our proposed formulae working with meshes composed of 3D ele-ments, using an inflate test of a rectangular cushion and a traction test of a partially perforated plate.
• Finally, we explore how the formulae works when using a complex law of behavior.

Equivalence between the formulation of Han and Lee and our pro-posals

The objective in this section is to verify the equivalent calculation between the method proposed by Hann and Lee, 3.10, modified by Troufflard [42] and our formulae 3.12 and 3.14. Notice that considering the term σxy in the formule 3.10 is problematic, because it depends on the coordinates on which it is calculated. In our work, in order to suppress this dependance, we propose to be situated in a stress eigenvector frame, which leads, considering the plain stress hypothesis, to σxy = 0, σx + σy = trace(σ) and γ = 3 in (3.10).

Table of contents :

1 Introduction 
1.1 Background and motivation
1.2 Industrial and academic objectives
1.3 Thesis overview
2 FE Software and validation 
2.1 FE software
2.1.1 Introduction
2.1.2 Algorithms
2.2 Validation
2.2.1 Geometry and calculation parameters
2.2.2 Analytical calculation
2.2.3 Herezh++ calculation and comparison with analytical results
2.2.4 Conclusions
3 Dynamic Relaxation Method 
3.1 Introduction
3.2 Numerical study of dynamic relaxation with kinetic damping applied to inflatable fabric structures with extensions for 3D solid element and nonlinear behavior
3.2.1 Introduction
3.2.2 Dynamic Relaxation Method
3.2.3 Numerical case studies
3.2.4 Conclusions and discussion
3.2.5 Acknowledgements
3.3 Comparison of several dynamic relaxation methods in the case of large displacements of membrane structures
3.3.1 Introduction
3.3.2 DR methods
3.3.3 Numerical case studies
3.3.4 Conclusions and discussion
3.3.5 Acknowledgements
4 3D measuring 
4.1 Introduction
4.2 Description of the devices used in the measuring
4.3 System calibration and sample preparation
4.4 Limitations and objectives of the experimentation
4.5 Numerical simulation
4.6 Measuring results
4.7 Conclusions
5 Tensile tests on fabric 
5.1 Introduction
5.2 Technic textile
5.3 Tensile test preparation
5.4 Results
5.5 Conclusions and discussion
6 Human body model 
6.1 Introduction
6.2 Software’s choice
6.3 Geometric considerations
6.4 Development in Gmsh
6.5 C++ programming and testing
6.6 Dummy’s design improving
6.6.1 General dimensions
6.6.2 Joints
6.7 Conclusions and discussion
7 Lifejacket modeling and contact mechanics 
7.1 Introduction
7.2 Creation and meshing of the lifejacket model
7.3 Numerical simulation of inflation
7.4 Contact mechanics
7.4.1 How contact is implemented
7.4.2 Numerical examples of contact (validation tests)
7.4.3 Application to the case of lifejacket and mannequin
7.5 Conclusions and discussion
8 Water modelling 
8.1 Introduction
8.2 Hydrostatic pressure
8.3 Hydrodynamic forces
8.4 Added mass and impact force
8.4.1 Impact force
8.5 Conclusions and discussion
9 General conclusion and perspectives 
9.1 Conclusions
9.2 Perspectives

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