Hamiltonian Particle Method Research Articles

Comparison of meshless discretizations in peridynamics and Hamiltonian particle method

Comparison of meshless discretizations in peridynamics and Hamiltonian particle method

An accuracy analysis of a Hamiltonian particle method with the staggered particles for seismic-wave modeling including surface topography

A Hamiltonian particle method (HPM), which is one of the mesh-free methods, can simulate seismic wavefields for models including surface topography in a simple manner. Numerical error caused by a curved free surface or by particles not aligned with the surface is not obvious in HPM. In general, the accommodation of irregular free surfaces requires more grids or particles in a minimum wavelength for achieving sufficient accuracy in the simulation. We tested the accuracy of HPM with staggered particles for simulating seismic-wave propagation including the surface topography, and we established the relationship between desired accuracy and spatial resolution. We conducted numerical simulations for models with a planar free surface aligned with the regular particle alignment and a dipping free surface. Our accuracy tests revealed that the numerical error strongly depends on the dipping angle of the slope. We concluded that about 25 particles in a minimum wavelength are required to calculate Rayleigh waves propagating along the irregular topography with good accuracy. Finally, we simulated Rayleigh wave propagation along irregular topography using a layered model with a hill. HPM can reproduce not only surface-wave propagation but also the reflected and refracted waves. Our numerical results were in good agreement with those from a finite-element method. Our investigations indicated that HPM could be a solution to simulate Rayleigh waves in the presence of complex surface topography.

Numerical simulation using a Hamiltonian particle method for effective elastic properties in cracked media

We apply a Hamiltonian particle method, one of the particle methods, to simulate seismic wave propagation in a cracked medium. In the particle method, traction free boundaries can be readily implemented and the spatial resolution can be chosen in an arbitrary manner. Utilisation of the method enables us to simulate seismic wave propagation in a cracked medium and to estimate effective elastic properties derived from the wave phenomena. These features of the particle method bring some advantages of numerical efficiencies (e.g. calculation time, computational memory) and the reduction of time for pre-processing.We describe first our strategy for the introduction of free surfaces inside a rock mass, i.e. cracks, and to refine the spatial resolution in an efficient way. We then model a 2D cracked medium which contains randomly distributed, randomly oriented, rectilinear, dry and non-intersecting cracks, and simulate the seismic wave propagation of P- and SV-plane waves through the region. We change the crack density in the cracked region and determine the effective velocity in the region. Our results show good agreement with the modified self-consistent theory, one of the effective medium theories. Finally, we investigate the influence of the ratio of crack length to particle spacing on the calculated effective velocities. The effective velocity obtained becomes almost constant when the ratio of crack length to particle spacing is more than ~20. Based on this result, we propose to use more than 20 particles per crack length.

A Hamiltonian Particle Method with a Staggered Particle Technique for Simulating Seismic Wave Propagation

We present a Hamiltonian particle method (HPM) with a staggered particle technique for simulating seismic wave propagation. In the conventional HPM, physical variables, such as particle displacement and stress, are defined at the center, i.e., at the same position, of each particle. As most seismic simulations using finite difference methods (FDM) are practiced with staggered grid techniques, we know the staggered alignment of space variables could improve the numerical accuracy. In the present study, we hypothesized that staggered technique could improve the numerical accuracy also in the HPM and tested the hypothesis. First, we conducted a plane wave analysis for the HPM with the staggered particles in order to verify the validity of our strategy. The comparison of grid dispersion in our strategy with that in the conventional one suggests that the accuracy would be improved dramatically by use of the staggered technique. It is also observed that the dispersion of waves is dependent on the propagation direction due to the difference in the average spacing of the neighboring two particles for the same parameters, as is usually observed in FDM with a rotated staggered grid. Next, we compared the results from the conventional Lamb’s problem using our HPM with those from an analytical approach in order to demonstrate the effectiveness of the staggered particle technique. Our results showed better agreement with the analytical solutions than those from HPM without the staggered particles. We conclude that the staggered particle technique would be a method to improve the calculation accuracy in the simulation of seismic wave propagation.

Numerical simulation of seismic wave propagation produced by earthquake by using a particle method

SUMMARY We propose a forward wavefield simulation based on a particle continuum model to simulate seismic waves travelling through a complex subsurface structure with arbitrary topography. The inclusion of arbitrary topography in the numerical simulation is a key issue not only for scientific interests but also for disaster prediction and mitigation purposes. In this study, a Hamiltonianparticlemethod(HPM)isemployed.Itiseasytointroducetraction-freeboundary conditionsinHPMandtorefinetheparticledensityinspace.Anymodelwithcomplexgeometry and velocity structure can be simulated by HPM because the connectivity between particles is easily calculated based on their relative positions and the free surfaces are automatically introduced. In addition, the spatial resolution of the simulation could be refined in a simple manner even in a relatively complex velocity structure with arbitrary surface topography. For these reasons, the present method possesses great potential for the simulation of strong ground motions. In this paper, we first investigate the dispersion property of HPM through a plane wave analysis. Next, we simulate surface wave propagation in an elastic half space, and compare the numerical results with analytical solutions. HPM is more dispersive than FDM, however, our local refinement technique shows accuracy improvements in a simple and effective manner. Next, we introduce an earthquake double-couple source in HPM and compare a simulated seismic waveform obtained with HPM with that computed with FDM to demonstrate the performance of the method. Furthermore, we simulate the surface wave propagation in a model with a surface of arbitrary topographical shape and compare with results computed with FEM. In each simulation, HPM shows good agreement with the reference solutions. Finally, we discuss the calculation costs of HPM including its accuracy.

地震波動伝播シミュレーションにおける粒子法の適用性に関する研究

地震国である我が国では，地震発生時の構造物の安全性や液状化対策などを考えることは，安全・安心な社会の実現にとって重要な課題である。地震防災を考える時，地震発生により生じる強震動を正確に予測することは，被害予測や社会基盤施設の復旧シミュレーションなどをおこなう上で最も基本的な情報となる。 従来から，地震波動伝播シミュレーションにはスタガード格子を用いた差分法が広く用いられてきた。また，自由境界条件の取り扱いの容易さなどから有限要素法の適用も試みられている。本研究では，粒子法の一種である Hamiltonian Particle Method (HPM) に着目し，地震波動伝播シミュレーションへの適用性を検討する。HPM は，有限要素法と同様に自由境界条件の取り扱いが容易であり，任意の地表面形状を有する解析対象を扱いやすい。また，離散化には粒子点座標を必要とするのみで，有限要素法のように節点と要素のコネクティビティを必要としないといった利点も有する。一方で，HPM はシンプレクティックスキームを適用することでエネルギー保存精度に優れた解析が可能であるが，粒子が局所的な振動を生じてしまい大域的な振動が減衰してしまうという問題がある。この問題を解決するために，振動抑制のための人工的な力の導入がおこなわれており，梁の自由振動解析などでその効果が検討されている。本研究では，地盤を伝播する地震波動のシミュレーションをする際に，この人工的な力がどのように結果に影響を及ぼすかに注目し，詳細な検討をおこなった。また，地震による強震動を考慮する際には，地表面を伝播する表面波を正確に再現できることは重要であるので，表面波の再現性を他の数値解析手法による結果と比較することで詳細に検討した。その結果，HPM を地震波動場のモデリングに適用する場合には，人工的な力の導入は不可欠であり，その大きさを規定する係数には媒質のヤング率の 0.5 倍程度の値を使用することが望ましいことがわかった。また，水平でない地表面を有するモデルにおける表面波伝播についてもシミュレーションをおこなったところ，表面波伝播に関しては実体波部分よりも数値的な振動が生じやすく，振源周波数や粒子間隔を適切に設定する必要があることがわかった。

Suppressing local particle oscillations in the Hamiltonian particle method for elasticity

AbstractThe governing equation of elasticity is discretized into motion equations of the particles in a Hamiltonian system. A weighted least‐square method is adopted to evaluate the Green–Lagrange strain. Using a symplectic scheme for the Hamiltonian system, we obtain the property of energy conservation in the discretized calculations. However, local particle oscillations occur, and they excessively decrease low frequency motion. In this study, we propose the use of an artificial potential force to suppress the local oscillations. The accuracy of the model with and without the inclusion of the artificial force is examined by analyzing a cantilever beam and wave propagation. With the inclusion of the artificial force, the local oscillations are reduced while energy conservation is maintained. Copyright © 2009 John Wiley & Sons, Ltd.

Numerical Simulation of Standing Waves Using the Hamiltonian Particle Method

The Hamiltonian particle method for incompressible fluid flows has been developed by authors and proved to have excellent properties in conservation of mechanical energy as well as linear and angular momenta. In this study, standing waves in a rectangular tank are simulated using this method. The result is compared to analytical solutions which are obtained from the Stokes perturbation expansion. Agreement between the numerical and the analytical solutions is fairly good at least in the initial two or three cycles. The numerical result is also compared to the one obtained using the moving-particle semi-implicit (MPS) method. The supremacy of the Hamiltonian particle method compared to the MPS method is confirmed.

A Hamiltonian particle method for non‐linear elastodynamics

AbstractParticle methods are meshless simulation techniques in which motion of continua is approximated by discrete dynamics of a finite number of particles. They have a great degree of flexibility, for instance, in dealing with complex large deformations or the fragmentation of solids. In this paper, a particle method for non‐linear elastodynamics of compressible and incompressible materials is developed based on a discretization of the Lagrangian from which the governing equations of elastodynamics are derived using the principle of least action. The discretized Lagrangian leads to a finite‐dimensional Hamiltonian system via the Legendre transformation. If the material is incompressible, the Hamiltonian system is accompanied by holonomic constraints. Depending on whether the material is compressible or incompressible, the symplectic scheme adopted for numerical time integration is either the Störmer/Verlet scheme or the RATTLE method, respectively. The resulting particle method inherits the symplectic structure possessed by the governing equations of elastodynamics. In the case of incompressible materials, incompressibility is strictly enforced at each time step. Some numerical tests indicate the excellence of the method for conservation of mechanical energy besides that of linear and angular momenta. Copyright © 2007 John Wiley & Sons, Ltd.

157 Hamiltonianに基づく超弾性粒子法に関する予備的検討(メッシュフリー/粒子法による大変形解析,OSO9 メッシュフリー/粒子法と関連技術)

157 Hamiltonianに基づく超弾性粒子法に関する予備的検討(メッシュフリー/粒子法による大変形解析,OSO9 メッシュフリー/粒子法と関連技術)

Hamiltonianに基づく粒子法の開発 : 粘性項の検討(OS8e メッシュフリー/粒子法)

Hamiltonianに基づく粒子法の開発 : 粘性項の検討(OS8e メッシュフリー/粒子法)