The top panel shows the acceleration in the x direction for one of the two affected particles. The middle panel shows the y-acceleration of the four particles in the top half of the target region.
Motivation
In this introduction, we describe the motivation for this work, our goals and achievements, provide a general description of granular crystals, followed by a brief historical overview of previous work in the field, and finally present the organization of this dissertation. Some of the most common methods currently used to dampen shock and vibration include damping (including but not limited to the use of polymer foams) [1], harmonic absorbers [2] and viscous dampers (such as in a typical door closer) between other examples.
Goals and achievements
Here, we performed systematic numerical simulations and experiments to investigate the effect of the material properties (elastic modulus and mass) of an interstitial spherical intruder placed at the center of a 2D square packing subjected to impact. The advantages and disadvantages of gradient methods over heuristics are discussed in [20].
Granular Crystals
Hertzian Contact and Nonlinearity
The nonlinearity in this case arises from the geometry of the contact and the dramatic increase of the contact area with the compression force. It is this intrinsic nonlinearity of the contact interaction that explains the nonlinearity of the granular crystals described in this thesis.
Dynamical Regimes and Example of the 1D Homogeneous Chain 7
MacKay published a rigorous mathematical proof of the existence of the predicted solitary wave [39] based on the existence theorem of Friesecke and Wattis for localized traveling waves in 1D lattices with nearest neighbor interactions [40]. Job and collaborators studied various aspects of the propagation of highly nonlinear solitary waves in 1D granular crystals.
Conceptual Organization of This Thesis
Square Packings, Equipartition Experiment and Hexagonal Pack-
- Experimental Setup 1: Square Packing
- Experimental Setup 2: Equipartition
- Experimental Setup 3: Hexagonal Packing
A first experimental setup was designed to investigate the propagation of stress waves in a 2D square packing of defect particles. Finally, a third experimental structure was designed and printed to assemble the hexagonal packings studied in Chapters 5 and 6.
Custom-Made Accelerometer Sensors
A stiff epoxy was used to glue the accelerometer into the drilled particle to maintain the bulk stiffness of the sensor particle. Photo of the accelerometer. right) Photograph of a steel sensor particle (accelerometer embedded in a drilled steel sphere).
Data Acquisition and Post-Processing Apparatus
In 2D experiments, we can access the acceleration at four locations simultaneously (each channel corresponds to the x or y component of the acceleration of the sensor particle). Accessing wave propagation data at more than four locations simultaneously (see for example Chapter 4) requires multiple runs to be performed with the same time reference.
Numerical Setups
Numerical Modeling and Integration Schemes
A spherical interstitial intruder is placed in the center of the packing and the system is impacted from the left by an impact particle.
Integration Schemes
R(u,u) =¨ Mu¨+S(u)−Fex= 0, (2.2) where R is the residual vector, M is the positive definite symmetric mass matrix, u and ¨u are the displacement and acceleration nodal vectors, Fext is the external nodal force vector and S(u) represents the internal nodal force vector. A time step of 1×10−6 s is sufficient to ensure conservation of the total energy with a similar accuracy to the Runge-Kutta solver.
Effect of Dissipation
As disorder increases, the system's propagation capacity becomes saturated and the wave becomes completely delocalized. It turns out that the stiffer the defect, the more it promotes the redistribution of energy in the system.
Experimental and Numerical Setups
After the interaction of the incoming solitary wave with the fault, transmitted, reflected and scattered solitary waves are visible. In both cases, the solid blue curve represents the displacement of the defect particle in the x direction, while the dashed green curve represents the displacement of the same particle in the y direction.
Single Defect
- Single Steel Defect
- Single PTFE Defect
- Relative Displacements
- Effect of Density
- Rigid Body Collision Model
The relative displacements of the defective particle with its neighbors are shown in detail in Fig. This translation is followed by small oscillations of the fault between its upper left and.
Two Defects
Two Defects in Contact with the Same Particle
This can be seen by comparing the displacement of the first indenter in the x-direction with the displacement of a single defect occupying the same spacing (see Figure 3.9). We calculate numerically how the input energy is redistributed across the different chains of the system.
Rigid Body Collision Model
We find that the corresponding analysis is instructive as a lower bound of the corresponding energy partition that satisfies the transmitted fraction of 87.17%. The colors and labels of the particles correspond to the colors and labels of the acceleration curves in panels 3.11(c) and 3.11(b).
Two Defects in a Line
An example is given in Fig.3.10 where we compare the waveforms of the acceleration signal in the "sticky" chain five particles after the intruder (solid red curve) and nine particles after the defect (black square markers). We notice that the shapes of the two signals (numerical solution and analytical solution) are very close.
Summmary
When placed closer to each other, the two defects begin to interact and the behavior of the second defect becomes more complicated, as its interaction with the transmitted (fast) wave and the scattered (slow) wave in the adjacent chain overlap. A soft defect particle spatially localizes a small percentage of the energy input to the crystal as it oscillates between its nearest neighbors.
Experimental Setups
This phenomenon is of interest in the creation of new acoustic waveguides, delay lines and strain-relieving materials. In the case of coupled Toda lattices, it was shown numerically that solitonic excitations applied to each of the coupled chains can result in the two different dynamical regimes (attractors) [127].
Numerical Setup
In both experiments, the system was impacted on one side by a steel impactor particle identical to those that made up the square packing, and custom-fabricated sensor particles were used to access the acceleration at each site of interest in the crystal (particles of red in Fig. 4.1). The sensor particles and data acquisition devices are described in detail in Sections 2.1.2 and 2.1.3 respectively.
Results and Discussion
Monomer Configuration
This corresponds to ≈4% of the kinetic energy, neglecting energy leakage in the transverse direction. Conversely, the amplitude of the wave propagating in the absorbing circuit increases from zero to the same level as the signal in the excited circuit.
Dimer Configuration
The black horizontal lines show the stabilized amplitude of the main acceleration pulse (≈ 800 m/s2 in numerical simulations and 400 m/s2 in experiments).
Simplified 1.5D Modelization and Comparison with Full 2D System . 67
In both graphs, the curves are obtained from the 2D model, while the (purple) stars indicate the peak amplitudes of the corresponding signals obtained from the 1.5D model. The second part of the work presented in this chapter consists of implementing a similar optimization scheme that optimally selects the material properties of each particle in the hexagonal packing.
Model
Optimization Problem
- General Formulation of the Optimization Problem
- Design Variable and Regularization Techniques
- Constraints
- Sensitivity Analysis
The evaluation of the response derivative terms Du(n), Dv(n) and Da(n) is more complicated due to the implicit dependence of the response variables on the design vector. In 5.2(b) we maximize the average energy reaching the two middle parts of the side walls.
Numerical Results
Topology Optimizations (Intruder Problem)
- Test Cases: Dispersion and Deflection Problems
- Other Examples: Layered Design, “3-Phase” Material
We plot the total peak energy (ie the maximum over the number of elements in the target area) as a function of time. The labeling and coloring of each curve correspond to the labeling and coloring of the box of the different configurations around the plot.
Material Problem
On the other hand, part of the incoming energy will be reflected at a light/heavy interface. We noticed that the optimizer performs better for low values of the initial design vector.
Experimental Results
Experimental Setups
The next section describes the experimental setups and presents experimental reproduction of some of the optimized designs. The top panel shows the acceleration in both the x and y directions for one of the two affected particles.
Comparison and Discussion
- Topology Optimizations (Intruder Problem)
- Material Problem
The upper panel shows the acceleration in the x direction for one of the two affected particles. The bottom panels show the acceleration in the y direction for the four particles in the upper half of the target area.
Conclusion
In both 5.14(b) and 5.14(c), each panel shows the acceleration of the corresponding labeled and colored particle in 5.14(a). In both 5.15(b) and 5.15(c), each panel shows the acceleration of the corresponding labeled and colored particle in 5.15(a).
Introduction
A Comparative Study of Material Optimization of Grain Crystals Using the Gradient Method and the Breeder Genetic Algorithm. Non-gradient-based (or heuristic) methods (see, among other examples, evolutionary computation of simulated annealing genetic algorithms) are well suited to problems where the solution space is multidimensional, multimodal, discontinuous, and noisy [159].
Model at Hand and Optimization Techniques
Numerical Modeling
In this chapter, we compare two approaches (BGA and gradient-based method), both used in previous studies for the optimal design of granular crystals [9]. More specifically, we compare the design and suitability obtained by performing material optimizations on 2D hexagonal gaskets.
Optimization Problem and Approaches
- Gradient-Based Method
- Breeder Genetic Algorithm
We are looking for discrete designs and xi is bound to take one of the two values xmin or xmax. To create the other λ−1 individuals of the new generation, parents are selected among the best µ individuals according to a uniform probability distribution, and recombined by means of an Extended Intermediate Recombination (EIT) scheme.
Optimized Designs and Dynamic Responses
Interpretation of the Optimized Designs
In each version, the black particles are made of steel and the white particles are made of PTFE. When maximizing the energy on the sides (see Figure 6.4), both solutions are very similar and place two lines of steel particles that directly connect the point of impact with the centers of the target areas where the energy is expected to be maximum.
Comparison and Discussion
Performance
This seems to indicate that most local optima of the design space have comparable performance to the global solution. The solid blue curve shows the response of the solution obtained via the BGA approach and the dotted green curve shows the response of the solution obtained via the gradient-based approach.
Computational Time
This should be compared to the number of transient and sensitivity analyzes performed in the gradient-based approach Niteration×(1 +M)T, where M is the number of design variables and constraints and in our study is typically equal to 0.004. The number of design variables N is equal to 33 for the material optimization problems we consider.
Summary
Daraio, “Internal energy localization through discrete gap breathing in one-dimensional diatomic granular crystals,” Phys. Vakakis, “Traveling waves and localized modes in one-dimensional homogeneous grain chains without precompression,” Phys.
