Dynamic Analysis

The dynamic analysis option permits two- and three-dimensional, fully dynamic analysis with MPoint. The dynamic capability applies to a wide range of problems in disciplines such as earthquake engineering, material processing and flow, or high-velocity impact. The calculation is based on the general formulation presented in the Theoretical Background). This Section describes the various features and considerations associated with the dynamic option in MPoint (i.e., dynamic loading and boundary conditions, wave transmission and mechanical damping).

Loading and Boundary Conditions

Users are encouraged to become familiar with the general concepts and Dynamic Modeling Considerations from the FLAC3D documentation, which are generally relevant for dynamic analyses in MPoint. In particular, dynamic loading and boundary conditions can be difficult to apply in MPoint since the MPM discretization does not explicit represent surfaces and boundaries. Capabilities such as Quiet Boundary Conditions are not available in MPoint.

For these reasons, it is recommended for certain dynamic analyses to use FLAC zones to represent the outer boundary of the model, apply base input motion and boundary conditions, and to use material points in regions where large deformation and material flow is expected. This is illustrated by the Landslide due to an Earthquake Example.

Note

A valid FLAC license with Dynamic option is required to couple zones together with material points and perform such simulations.

The dynamic feature can be coupled to the groundwater flow model (currently limited to undrained conditions). This allows, for example, analyses involving time-dependent pore pressure change associated with liquefaction.

Dynamic Damping

Non-viscous local and combined damping, hysteretic damping, and viscous Rayleigh and Maxwell damping are available in MPoint. Damping is prescribed to the entire domain using the mpoint node dynamic command.

Local and Combined Damping

The non-viscous local and combined damping force is applied to the MPM background grid nodes, where the equations of motion are solved. Momentum Equation (16) from the Theoretical Background for a node \(i\) becomes:

(1)\[\begin{equation} m_{i} \mathbf{a}_i = \mathbf{f}_i + \mathbf{f}_i^{\text{damp}} \end{equation}\]

where \(\mathbf{f}_i = \mathbf{f}_i^{\text{ext}} + \mathbf{f}_i^{\text{int}}\). The damping force \(\mathbf{f}_i^{\text{damp}}\), for each degree of freedom \(<l>\), is computed as follows for the local damping:

(2)\[\begin{equation} \mathbf{f}_i^{\text{damp, local} <l>} = - \alpha \left| \mathbf{f}_i^{<l>} \right| \mathrm{sign} (\mathbf{v}_i^{<l>}) \end{equation}\]

and for the combined damping:

(3)\[\begin{equation} \mathbf{f}_i^{\text{damp, combined} <l>} = \frac{\alpha \left| \mathbf{f}_i^{<l>} \right|}{2} \left[ \mathrm{sign}\left( \frac{d\mathbf{f}_i^{<l>}}{dt} \right) - \mathrm{sign}\left(\mathbf{v}_i^{<l>}\right) \right] \end{equation}\]

Rayleigh Damping

Rayleigh damping generally contains a mass-proportional and a stiffness-proportional component. Similar to the FLAC implementaion described Here, the stiffness-proportional component is introduced as a viscous stress increment, while the mass-proportional component modifies the velocity update.

Because of the double discretization in MPM, the stress calculation for the stiffness-proportional component is performed on the material point, while the modification of the velocity is performed on the background grid nodes. The viscous stress increment \(\Delta \pmb{\sigma}_p^{\text{visc}}\) at the material point is computed as:

(4)\[\Delta \pmb{\sigma}_p^{\text{visc}} = \frac{\beta}{\Delta t} \Delta{\pmb{\sigma}}_p\]

from the stress increment \(\Delta \pmb{\sigma}_p\) returned from the constitutive model, and used to compute the background grid damping force as:

(5)\[\begin{equation} \mathbf{f}_i^{\text{damp, Rayleigh}} = - \sum_{p=1}^{n_p} V_p \Delta \pmb{\sigma}_p^{\text{visc}} \cdot {\nabla} w_{ip} \end{equation}\]

The mass-proportional component of Rayleigh damping modifies the velocity update of the background node to:

(6)\[\tilde{\mathbf{v}}_i^{n+1/2} = \frac{\left(1 - \alpha \frac{\overline{\Delta t^n}}{2}\right) \mathbf{v}_i^{n-1/2} + \overline{\Delta t^n}\mathbf{a}_i^n}{1 + \alpha \frac{\overline{\Delta t^n}}{2}}\]

where the background node acceleration \(\mathbf{a}_i^n\) includes the contribution of the Rayleigh viscous stress increment described above according to (1).

Maxwell Damping

Maxwell damping is implemented using the implicit algorithm described here.

For each Maxwell component \(k\), with relaxation time \(\tau_k\), the Maxwell stress \(\pmb{\sigma}_{p, k}\) at the material point is updated from the stress increment \(\Delta \pmb{\sigma}_p\) returned from the constitutive model as:

(7)\[\pmb{\sigma}_{p, k}^{n+1} = \frac{\alpha \Delta \pmb{\sigma}_p + \pmb{\sigma}_{p, k}^n}{1 + \frac{\Delta t}{\tau_k}}\]

and similar to Rayleigh damping, the damping nodal force is computed as:

(8)\[\begin{equation} \mathbf{f}_i^{\text{damp, Maxwell}} = - \sum_{p=1}^{n_p} V_p \left(\sum_{k=1}^{3}\pmb{\sigma}_{p, k}\right) \cdot {\nabla} w_{ip} \end{equation}\]

Note

A valid FLAC license with Dynamic option is currently required to use Maxwell damping. A standalone MPoint capability is under development.

Hysteretic Damping

The hysteretic damping applies a reduction factor to the shear modulus during constitutive model calculation at the material point. The calculation of this reduction factor is detailed Here.

Note

A valid FLAC license with Dynamic option is currently required to use Hysteretic damping. A standalone MPoint capability is under development.