Mathematical Physics - Volume II - Numerical Methods

5.10 Zero-Energy Modes

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The stability analysis of the Eulerian SPH equations, presented above, has revealed that the stability of the system is governed by three terms: a material stability term, which is desirable since this term is also present in the continuum equations, and two more terms which are the result of the type of discretisation carried out, namely the spurious singular mode term and the tensile instability term. An effective illustration of the distortion of material instability, by the spatial discreti sation, for the case of a hyperelastic material is given in Figure 5.5 taken from [73]. The domains of material stability for the Lagrangian kernel, an Eulerian kernel and that of the governing partial differential equation, i.e. the momentum equation are clearly identified. The stable domains are defined in the space of the two principal stretches, λ 1 and λ 2 .

Figure 5.5: (a) Stable domain for MLS particle method with stress point integration and Lagrangian kernel compared to the stable domain for the PDE; (b) Stable domains of MLS particle methods for stress point integration with Eulerian and Lagrangian kernel for hyperelastic material; dashed and solid lines bound the stable domains for Lagrangian and Eulerian kernels, respectively [73].

5.10 Zero-Energy Modes Zero-energy modes are a problem that is not unique to particle methods. These spurious modes, which correspond to modes of deformation characterised by a pattern of nodal displacement that produces zero strain energy, can also be found in the finite difference and finite element methods. Swegle [87] was first to show that SPH suffers from zero energy modes. These modes arise from the nodal under integration. The fundamental cause is that all field variables and their derivatives are calculated at the same locations (particle positions), which makes the SPH method collocational. For instance, for a 1D oscillatory velocity field, illustrated in Figure 5.6, the kernel approximation would give negligible velocity gradients and consequently stresses at the particles. These modes of deformation are not resisted and can be easily exited by rapid impulsive loading. Another explanation can be found in the origin of the kernel approximation. As the kernel approximation, which is the basis of