# A local adaptive discretization algorithm for Smoothed Particle Hydrodynamics

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## Abstract

In this paper, an extension to the Smoothed Particle Hydrodynamics (SPH) method is proposed that allows for an adaptation of the discretization level of a simulated continuum at runtime. By combining a local adaptive refinement technique with a newly developed coarsening algorithm, one is able to improve the accuracy of the simulation results while reducing the required computational cost at the same time. For this purpose, the number of particles is, on the one hand, adaptively increased in critical areas of a simulation model. Typically, these are areas that show a relatively low particle density and high gradients in stress or temperature. On the other hand, the number of SPH particles is decreased for domains with a high particle density and low gradients. Besides a brief introduction to the basic principle of the SPH discretization method, the extensions to the original formulation providing such a local adaptive refinement and coarsening of the modeled structure are presented in this paper. After having introduced its theoretical background, the applicability of the enhanced formulation, as well as the benefit gained from the adaptive model discretization, is demonstrated in the context of four different simulation scenarios focusing on solid continua. While presenting the results found for these examples, several properties of the proposed adaptive technique are discussed, e.g. the conservation of momentum as well as the existing correlation between the chosen refinement and coarsening patterns and the observed quality of the results.

## Keywords

Particle simulation Smoothed Particle Hydrodynamics Model discretization Adaptivity Refinement Coarsening## 1 Introduction

Due to its meshless nature, typical fields of application of the Lagrangian particle method Smoothed Particle Hydrodynamics (SPH) are simulation scenarios that are characterized by large deformations of the modeled domain of material, disintegrating structures, and/or highly dynamic impacts. But although there are no issues coming up that are a result of a distorted mesh when employing a meshless discretization method, the model quality is still highly dependent on the number of interpolation points that are used for the discretization of the investigated spatial domain. For this reason, an important field of research concerning SPH is the development of discretization strategies that allow for an adaptation of the number of SPH particles being placed in certain areas of a simulation model at runtime. Recent research activities dealt with fundamental issues of such adaptive extensions and provided the basis for some first combined refinement and coarsening techniques. The approach proposed in [1] gives a good starting point, but relies on an exclusively random-based particle selection strategy for the coarsening process, which is restricted to pairs of adaptive particles. Therefore, we have been working on a more advanced adaptive discretization extension to the SPH method over the last few years.

After having developed an enhanced version of the refinement procedure presented in [2] and [3], as well as an appropriate coarsening algorithm, we implemented the combined adaptive discretization strategy into the SPH plugin for the particle simulation package Pasimodo [4]. By testing the approach for a broad range of different simulation setups, we have been able to verify its validity. In this paper, the theoretical basis of this newly developed discretization approach, along with some of the validation examples, is presented.

First, a brief introduction to the SPH spatial discretization method in general as well as to its most important extensions necessary to be able to reproduce the behavior found for real solid matter is provided.

Second, the modifications made to allow for a local adaptive refinement and coarsening of the modeled structure are presented. In this context, suitable criteria to decide whether the discretization level of an existing particle is to be adapted or not are discussed. In case of refinement, different patterns specifying the positions of the additional particles are introduced and, in case of coarsening, different strategies to identify the particles to be merged are presented. Furthermore, the way how to determine the state variables of the new particles are discussed in detail. All of these modifications can also be applied to SPH fluid simulations without further adjustments.

Third and finally, four of the various simulation scenarios used for the validation of the implemented routine are presented. That way, it is shown that the introduced approach provides correct results when being applied to simulations focusing on solid continua. In a first step, the focus is laid on the basic properties of the presented adaptive technique, e.g. the conservation of kinetic energy. To that end, the results found for the simulations of an ordinary and a notched tensile test are discussed. For these two rather simple setups, analytical solutions as well as numerical results obtained with the Finite Element Method (FEM) [5] will be compared with the ones provided by the adaptive SPH method. In a second step, the applicability of the proposed approach to more typical and demanding scenarios is demonstrated for two examples; a rigid torus falling onto an elastic membrane and an orthogonal cutting process.

## 2 Smoothed Particle Hydrodynamics

Originally developed for the purpose of investigating astrophysical phenomena [6, 7], the meshfree Smoothed Particle Hydrodynamics (SPH) method experienced the necessary modifications to be also applicable to fluid simulations, i.e., to allow for a discretization of the well-known Navier-Stokes Eq. [8]. In this context, SPH, as a spatial discretization method, is used to replace the considered continuum domain with a set of particles, which can be interpreted as interpolation points. At these points, the underlying differential equations of continuum mechanics are evaluated. In doing so, the initial system of partial differential equations (PDEs) in time and space is transformed into one that consists only of ordinary differential equations (ODEs) in time [9], which can be numerically integrated with common integration schemes. In addition to fluid simulations, SPH has proven its ability to correctly reproduce the physical effects that are observed for real solid material over the last two decades [10, 11].

Next, the basic principle of the SPH discretization method is presented. After having described its basic idea, the SPH simulation technique is applied to the Euler equations, which can be used to reproduce the behavior of solid matter in simulations. Then, the focus is laid on some of the extensions to the basic SPH solid formulation introduced in [12] that improve the significance of the obtained results. Namely, these are the Johnson-Cook (JC) plasticity model, the JC damage model, as well as a modified version of the Lennard-Jones potential.

### 2.1 Basic principle of Smoothed Particle Hydrodynamics

The SPH discretization process to transform a PDE into an ODE consists of the following three steps. First, an ensemble of abstract, not discrete particles is introduced into the considered simulation domain. Each of these SPH particles represents a specific part of the original spatial domain and, according to that, combines all of its properties, such as position, mass, density, etc. That way, it is possible to eliminate the dependency on the spatial position and its derivatives in case of any PDE while using the SPH particles, i.e. the introduced interpolation points, in conjunction with the Dirac delta function.

Second, the Dirac delta function used for the discretization of a PDE is approximated by a kernel function of finite extent, e.g. a truncated Gaussian distribution. The contribution of each neighboring particle to a property of the particle of interest is weighted according to the distance between these two SPH particles as well as the actual shape of the chosen kernel function. For this reason, the kernel function is also referred to as weight function. That way, the level of smoothing of the properties of an SPH particle is determined. Commonly, the functions that are used as kernel functions are cut off at a user-defined distance from the interpolation center, i.e. the so-called smoothing length, to save computational effort by not taking into account the relatively minor contributions from distant particles.

Third, when using a finite number of SPH particles for the discretization of a spatial domain, the integral over the entire support area, which has been introduced in combination with the Dirac delta function in the first step, has to be replaced with a sum over all neighboring particles.

### 2.2 Discretized Euler equations

### 2.3 Plasticity model

### 2.4 Damage model

### 2.5 Force model for boundary interaction

## 3 Refinement and coarsening algorithm for Smoothed Particle Hydrodynamics

Many different technical processes of high industrial importance are characterized by a major increase in the level of plastic deformation, equivalent stress, or temperature that are restricted to small areas of the processed solid material. This leads to fairly inhomogeneous distributions of the mentioned state variables over the workpiece structure. Therefore, a finer discretization for these particular parts of the workpiece model is needed in order to achieve the same accuracy of the numerical results as found for the areas that are influenced only to a certain extent by the actual technical process. But instead of increasing the discretization level of the whole structure and with it the overall computational cost, the adaptation shall be restricted to the parts that really have need for it. By developing a combined local adaptive refinement and coarsening algorithm, we found an SPH formulation that allows for such an adaptive increase in the discretization level for specific areas of the model that satisfy given refinement criteria. At the same time, the number of SPH particles is decreased in domains where the criteria chosen for coarsening are fulfilled.

The adaptive refinement and coarsening approach presented in this section consists of three steps. First, it is decided if a particle needs to be adapted by checking whether a user-defined criterion is met or not. In the implementation, this can be a check of an absolute value, a gradient, and/or a relative geometric position. Second, if the criterion is met, the particle is split into several smaller ones or merged with some of its neighbors into a bigger one according to a chosen refinement or coarsening pattern. Third, the state variables of the new, adapted particles are calculated either directly from the corresponding values of the original particles or by interpolating them from the values provided by their neighboring particles. In the following two subsections, the refinement and coarsening approaches are presented separately, starting with the refinement algorithm.

### 3.1 Local adaptive refinement extension

The adaptive refinement approach presented in this section follows the procedure introduced in [2] and [3]. When using it, the first question to be answered is when to refine an SPH particle at all. One or several suitable criteria need to be chosen for each simulation setup, e.g. single particle values, field gradients over the neighborhood, or numerical data. While developing and testing the refinement algorithm, it turned out that for lots of different simulation scenarios, already quite simple refinement criteria that compare a single particle value, e.g. one of the particle’s spatial coordinates or its von Mises stress, to a given threshold work very well. Especially when applying the introduced refinement approach to avoid a numerical fracture, a criterion that is based on a particle’s coordination number, i.e. the overall number of neighbors that are located within its smoothing area, proved to be quite useful in this context. As we allow for multiple refinement, it is necessary to restrict the particle size or the number of times a particle is split up in addition to the user-defined refinement criteria. Otherwise, the number of particles could increase uncontrolled, leading to an enormous computational effort, or the refinement process could create particles with almost zero masses and smoothing lengths. This results in various numerical problems, such as very small time steps.

Up to now, the positions, masses, and smoothing lengths of the new particles have been specified. The other properties, such as the densities and the velocities, need to be set, too. While most state variables, as for example the densities, can be calculated via SPH interpolation from the ones of the original particles without experiencing any issues concerning a further inaccuracy of the simulation results, this is not the case for the velocities of the additional particles. When they are set to the ones of the original particles, a conservation of linear and angular momentum as well as kinetic energy is achieved. Choosing \(\varvec{v}_\mathrm {j} = \varvec{v}_\mathrm {i}\) for all particles \(j\) is the only way to ensure an exact conservation, which is a must when looking for reliable numerical results. Conversely, due to the fact that the smaller particles are positioned somewhere around the location of the original one without modifying their velocity values accordingly when following this approach, this choice may modify the existing velocity field and, thus, also lead to additional, unwanted inaccuracies in the simulation results. This dilemma needs to be further investigated in the future.

### 3.2 Local adaptive coarsening extension

The coarsening approach presented in this paper basically follows the same procedure as the adaptive refinement algorithm described above. First, those areas that are of little interest are to be identified. For this purpose, coarsening criteria need to be defined. These criteria can be chosen similar to the refinement routine, e.g. based on a particle’s velocity or its von Mises stress.

Originally, the aforementioned procedures were developed for the simulation of fluids. When considering SPH simulations that have their focus on solid material, the particles often experience a lot less relative motion and reordering than in case of fluid simulations. Inspired by the approach presented in [19], which keeps the original particles instead of removing them, we implemented a family-based coarsening strategy, which aims on merging particles that originate from the same particle if possible. For this purpose, each refined particle is initialized with a variable providing the ID of its origin particle. By adding an additional merging condition that ensures that SPH particles with the same origin are preferred for merging, one gets a suitable extension to the neighbor-based strategy that takes into account the refinement history of an adaptive particle.

## 4 Examples

Parameter values for the combined refinement and coarsening algorithm

Parameter | value |
---|---|

Allowed number of refinement steps per particle | 1 |

Refinement pattern (two- and three-dimensional) | Cubical |

Dispersion of refinement pattern \(\varepsilon \) | 0.5 |

Smoothing length multiplication factor \(\alpha \) | 0.85 |

Strategy for velocity calculation | \(\varvec{v}_\mathrm {j} = \varvec{v}_\mathrm {i}\) |

Allowed number of coarsening steps per particle | 1 |

Coarsening pattern (two- and three-dimensional) | Family-based |

Desired number of particles to be merged \(N\) (two- / three-dimensional) | 4 / 8 |

Tolerance limit for Newton-Raphson method | \(1 \cdot 10^{-6}\) |

Maximum number of steps for Newton-Raphson method | 20 |

Minimum number of time steps since last refinement | 5 |

### 4.1 Ordinary tensile test

The uniaxial tensile test is one of the most basic experimental setups that are employed to investigate the elastic, the plastic, and the fracture behavior of real solid material and is, for this reason, also suitable for a comparison with simulation data. It is the first step in our working process of transferring an ordinary flat tensile specimen used in experiments into a simulation model to build up a regular lattice of SPH particles that has the same shape and dimensions as the gauge section of the real specimen. For building up this lattice, the particles are positioned according to a structure consisting of quadratic cells. In addition to that, to both ends of the simulated specimen whose normal vector is parallel to the loading direction additional rows of SPH particles, which represent the grips applied to the real one, are attached. The particles belonging to one of these grip areas are fixed in space, while a constant velocity is applied to the particles of the other one. Thus, one finds an invariable elongation rate level throughout the entire simulation. The SPH particles that are located within the measuring area are not at rest at the beginning of the simulation, but to them an initial velocity according to their relative distance from the grip sections is assigned. That way, a linear velocity field is established in order to avoid having shock waves running from one end of the specimen to the other resulting in a noisy stress-strain-curve.

When applying the SPH simulation technique in its original formulation, i.e. without any adaptive extension, to the tensile test simulation or, speaking more generally, to problems showing large deformations of the modeled structure, a possible increase in the distance between adjacent particles can lead to a decrease in the number of neighbors that are located within the smoothing areas. Sooner or later, this results in a numerically induced fracture. To ensure that the damage behavior of the simulated material is controlled exclusively by the fracture model introduced in Sect. 2.4, even in case of large strain values, the presented refinement algorithm can be used to avoid a numerical fracture by adapting the model discretization in accordance to its deformation.

### 4.2 Notched tensile test

A more challenging evaluation setup for the introduced adaptive SPH formulation than the ordinary tensile specimen considered above is the simulation of a notched specimen due to the fairly inhomogeneous stress distribution found in this case. In analogy to the tensile specimen regarded in the previous section, a regular lattice with quadratic cells of SPH particles serves as a basis for the two-dimensional model of a flat double-edge notched tensile specimen with an initial width of \(1 \cdot 10^{-2} \, \hbox {m}\) and an initial length of \(1.34 \cdot 10^{-2} \, \hbox {m}\). Halfway up the length of the created geometry round notches with a radius \(r = 1.5 \cdot 10^{-3} \, \hbox {m}\) and an offset of \(2 \cdot 10^{-4} \, \hbox {m}\) from the outer edge are introduced on both long sides of the specimen. To achieve a satisfactory level of reproduction of the high gradients in stress that can be observed next to the roots of the notches, an initial particle spacing of \(2.3 \cdot 10^{-3} \, \hbox {m}\) is used for the discretization of the adaptive SPH simulation, leading to an overall number of particles of about \(2\,700\) at the beginning of the simulation. The cubical refinement pattern as depicted in Fig. 3 with \(\varepsilon = 0.5\) and \(\alpha = 0.85\) is employed and the chosen refinement criterion is stress-based. For comparison, an unrefined SPH simulation with an initial particle spacing of \(1.15 \cdot 10^{-3} \, \hbox {m}\) is performed, leading to an overall number of particles of about 10 000. As it is hardly possible to determine the stress distribution that can be found for a notched tensile specimen in an analytical way, two simulation-based references are used for comparing the results computed by the SPH method; a simulation of a notched tensile test performed by the Finite Element program ADINA and one employing the Discrete Element Method (DEM) [20].

### 4.3 Torus falling onto a membrane

In the initial configuration of the simulation setup, the torus is placed at some distance above the membrane. After having released the torus from its starting position, the gravitational force acting on it leads to a free fall motion of the rigid body until it hits the elastic membrane. With this contact, the process of transferring kinetic energy from the torus to the membrane begins. Subsequently, the rigid body is gradually decelerated until its vertical velocity completely vanishes. Due to the interparticle bonds, the amount of kinetic energy absorbed by the adaptive SPH particles is step-by-step transformed into elastic deformation energy. For this reason, the stress level of the particles is increased and they are refined once the employed stress-based refinement criterion is met. As can be seen in Fig. 12b, this first happens for the particles being directly in contact with the meshed body and, as a consequence, the adaptive discretization algorithm projects the geometry of the torus (dashed lines) onto the uppermost layer of SPH particles, i.e. the surface of the elastic membrane.

In the second part of the simulation, some of the energy being stored in the elastic membrane is retransferred to the torus as part of the rebound effect. Therefore, the torus does not stay at rest, but is accelerated in vertical direction and begins an upward motion. When the torus and the surface layer of the membrane are no longer in contact, the stress level of the uppermost membrane particles falls below the specified coarsening threshold value and, as shown in Fig. 12c, the previously refined particles are coarsened again.

### 4.4 Orthogonal cutting process

A simulation scenario that is more related to real technical applications than the previous examples is the orthogonal cutting process. The orthogonal cutting setup represents an elementary cutting process where the motion of the tool is perpendicular to the cutting edge. When assuming a sufficiently large width of cut compared to the chosen depth of cut, a two-dimensional state of stress can be assumed in the workpiece and the machining process becomes a planar problem [22].

Material properties for steel C45E at room temperature

Parameter | Value |
---|---|

Density \(\rho \) | \(7\,700 \, \hbox {kg/m}^3\) |

Young’s modulus \(E\) | \(2.1 \cdot 10^{11} \, \hbox {N/m}^2\) |

Poisson ratio \(\nu \) | 0.3 |

Yield strength \(\sigma _\mathrm {y}\) | \(7.11 \cdot 10^{8} \, \hbox {N/m}^2\) |

Fracture strain \(\varepsilon _\mathrm {F}\) | 0.133 |

Besides the workpiece material, the geometry of the cutting tool can have significant influence on the behavior of the cutting system and, therefore, the obtained results. Due to manufacturing reasons, the cutting edge of a real tool reveals deviations in its geometry from an ideal wedge shape [23]. As a consequence, not a tool with an ideal geometry, i.e. a wedge shape, but a more sophisticated one taking into account the imperfections resulting from limitations of the manufacturing process is employed for the presented machining simulation. One model that promises a sufficient imitation of a real tool is introduced in [24]. For the calculation of the particle forces resulting from the interaction with the cutting tool, once more the penalty approach presented in Sect. 2.5 is deployed. Further information on the presented orthogonal cutting setup can be found in [25]. The behavior of the cutting model that can be observed when using both the original as well as the adaptive SPH formulation is discussed hereinafter.

If the introduced adaptive discretization technique is applied to the orthogonal cutting simulation, one observes a refinement of the load-carrying particles in the different shear zones of the cutting specimen as it is shown in Fig. 13b. While the blue-colored particles still have the initial size, the red-colored ones have been refined in order to be able to resolve the high gradients in stress found for these regions. Again, a stress-based refinement criterion together with the cubical refinement pattern introduced in Sect. 3.1 is used. For the areas of the workpiece model that have already been passed by the tool geometry, the previously refined SPH particles are adaptively coarsened, as indicated by their red color, when showing a stress level that is at least ten times lower than the specified refinement threshold value.

## 5 Conclusion

In this paper, a local adaptive discretization algorithm for the SPH method has been proposed. With the presented approach, it is possible to improve the accuracy of the simulation results while reducing the required computational cost at the same time. To that end, the number of SPH particles is adaptively increased in certain areas of a simulation model if necessary, whereas it is decreased for domains of low interest. In a first step towards such a combined refinement and coarsening strategy, an enhanced version of the refinement procedure has been integrated into the SPH solid formulation that had been extended to the JC plasticity model, the JC damage model, as well as an appropriate boundary force model. In a second step, we developed a coarsening algorithm that is compatible with the refinement routine implemented before. Both adaptive components follow the same basic procedure, which consists of the following three steps. First, it is decided if the discretization level of an already existing SPH particle is to be adapted by checking whether a user-defined criterion is met or not. Second, if the criterion is met, the particle is split into several smaller ones or, respectively, merged with some of its neighbors into a bigger one according to a chosen refinement or coarsening pattern. Third, the state variables of the additional particles are calculated either by interpolation or by cloning and subsequent scaling the ones of the original particle. In a last step, the validity of the coupled approach, as well as its applicability, was shown by applying it to a broad range of different simulation setups.

Four of the various simulation scenarios used for the validation of the implemented adaptive discretization routine were then presented. In the context of an ordinary tensile test setup, the capability of the introduced adaptive discretization technique to avoid a numerical fracture as well as to ensure a conservation of linear and angular momentum as well as kinetic energy was demonstrated. Next, it was shown for a notched tensile test setup that the combined refinement and coarsening strategy provides the same results as found with the FEM or a lot more computational expensive SPH simulations with a fixed number of particles. The fact that the results quality of the adaptive SPH formulation depends on the choice which of the introduced refinement and coarsening patterns are employed was discussed. Besides these two simulation scenarios used to illustrate the basic qualities of the introduced local adaptive discretization algorithm, it is also applicable to SPH simulations showing a highly dynamic behavior of the modeled structures and is suitable to solve problems related to real technical applications. This has been shown in the context of a simulation of a rigid torus falling onto an elastic membrane and one of an orthogonal cutting process.

## Notes

### Acknowledgments

The research leading to the presented results has received funding from the German Research Foundation (DFG) under the Priority Program SPP 1480 grant EB 195/12-2. This financial support is highly appreciated.

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