Shift boundary material point method: an image-to-simulation workflow for solids of complex geometries undergoing large deformation


We introduce a mathematical framework designed to enable a simple image-to-simulation workflow for solids of complex geometries in the geometrically nonlinear regime. While the material point method is used to circumvent the mesh distortion issues commonly exhibited in Lagrangian meshes, a shifted domain technique originated from Main and Scovazzi (J Comput Phys 372:972–995, 2018) is used to represent the boundary conditions implicitly via a level set or signed distance function. Consequently, this method completely bypasses the need to generate high-quality conformal mesh to represent complex geometries and therefore allows modelers to select the space of the interpolation function without the constraints due to the geometric need. This important simplification enables us to simulate deformation of complex geometries inferred from voxel images. Verification examples on deformable body subjected to finite rotation have shown that the new shifted domain material point method is able to generate frame-indifferent results. Meanwhile, simulations using micro-CT images of a Hostun sand have demonstrated that this method is able to reproduce the quasi-brittle damage mechanisms of single grain without the excessively concentrated nodes commonly displayed in conformal meshes that represent 3D objects with local fine details.

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This paper is dedicated to Professor Jiun-Shyan Chen on the occasion of his 60th birthday. This work is supported by the Earth Materials and Processes program from the US Army Research Office under grant contract W911NF-18-2-0306, the Dynamic Materials and Interactions Program from the Air Force Office of Scientific Research under grant contract FA9550-17-1-0169, the nuclear energy university program from Department of Energy under grant contract DE-NE0008534 and the Mechanics of Material program at National Science Foundation under grant contract CMMI-1462760. These supports are gratefully acknowledged. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the sponsors, including the Army Research Laboratory or the US Government. The US Government is authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright notation herein.

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We linearize all the terms varying with \(\varvec{u}\) in (12) to implement the Newton–Raphson iteration. We here only show the procedure of linearization of the terms containing \(\varvec{\sigma } = \varvec{\sigma ({\varvec{u}}})\). There are two main approaches to linearize the weak form of the balance of linear momentum. One is to derive the rate of internal energy using an objective stress rate, in which the tangent stiffness matrix is composed by material tangent stiffness and geometric stiffness, as shown in [4]. In [12], the measurement of stress, such as \(\varvec{P}\), is a function of the displacement through the deformation gradient:

$$\begin{aligned} \varvec{P} = \varvec{P} \left( \varvec{F} \left( \varvec{u} \right) \right) . \end{aligned}$$

The directional derivative of \(\varvec{P}\) at \(\varvec{u}^*\) therefore is:

$$\begin{aligned} D \varvec{P}_{\varvec{u}= \varvec{u}^*} \cdot \delta {\varvec{u}} = \left. \frac{\hbox {d}}{\hbox {d}\epsilon } \right| _{\epsilon =0} \frac{\partial \varvec{P}}{\partial \varvec{F}} \frac{\partial \varvec{F} (\varvec{u}^* + \epsilon \delta \varvec{u} )}{\partial \epsilon } = \varvec{A} : {{\,\mathrm{\nabla ^{\varvec{X}}}\,}}{\delta \varvec{u}}. \end{aligned}$$

To some extent, the tangent modulus \(\varvec{A} = \frac{\partial \varvec{P}}{\partial \varvec{F}}\) is in a general form. We can also obtain

$$\begin{aligned} D \left[ \int _{\mathcal {B}_0} \varvec{\varvec{P}}:{{\,\mathrm{\nabla ^{\varvec{X}}}\,}}{\varvec{\eta }} \hbox {d}V\right] \cdot \delta \varvec{u}= & {} \int _{\mathcal {B}_0} \varvec{A} : {{\,\mathrm{\nabla ^{\varvec{X}}}\,}}{\delta \varvec{u}} : {{\,\mathrm{\nabla ^{\varvec{X}}}\,}}{\varvec{\eta }} \hbox {d}V, \nonumber \\\end{aligned}$$
$$\begin{aligned} D \left[ \int _{\Gamma _0} \varvec{\eta } \cdot \varvec{\varvec{P}} \cdot \varvec{N} \hbox {d}A \right] \cdot \delta \varvec{u}= & {} \int _{\Gamma _0} \varvec{\eta } \cdot \varvec{A}: {{\,\mathrm{\nabla ^{\varvec{X}}}\,}}{\delta \varvec{u}} \cdot \varvec{N} \hbox {d}A.\nonumber \\ \end{aligned}$$

Since the material and spatial virtual work functional are equivalent, we can transform (55) and (56) into

$$\begin{aligned} D \left[ \int _{\mathcal {B}} \varvec{\sigma } :{{\,\mathrm{\nabla ^{\varvec{x}}}\,}}{\varvec{\eta }} \hbox {d}V\right] \cdot \delta \varvec{u} =&D \left[ \int _{\mathcal {B}_0} \varvec{\varvec{P}} :{{\,\mathrm{\nabla ^{\varvec{X}}}\,}}{\varvec{\eta }} \hbox {d}V\right] \cdot \delta \varvec{u}\nonumber \\ =&\int _{\mathcal {B}} \frac{1}{J} \varvec{A} : ({{\,\mathrm{\nabla ^{\varvec{x}}}\,}}{\delta \varvec{u}} \varvec{F}): ({{\,\mathrm{\nabla ^{\varvec{x}}}\,}}{\varvec{\eta }} \varvec{F}) \hbox {d}V \nonumber \\ =&\int _{\mathcal {B}} \varvec{a} : ({{\,\mathrm{\nabla ^{\varvec{x}}}\,}}{\delta \varvec{u}} ) : ({{\,\mathrm{\nabla ^{\varvec{x}}}\,}}{\varvec{\eta }}) \hbox {d}V, \end{aligned}$$
$$\begin{aligned} D \left[ \int _{\Gamma } \varvec{\eta } \cdot \varvec{\sigma } \cdot \varvec{n} \hbox {d}A \right] \cdot \delta \varvec{u} =&D \left[ \int _{\Gamma _0} \varvec{\eta } \cdot \varvec{\varvec{P}} \cdot \varvec{N} \hbox {d}A \right] \cdot \delta \varvec{u}\nonumber \\ =&\int _{\Gamma _0} \varvec{\eta } \cdot \varvec{A}: {{\,\mathrm{\nabla ^{\varvec{X}}}\,}}{\delta \varvec{u}} \cdot \varvec{N} \hbox {d}A \nonumber \\ =&\int _{\Gamma } \frac{1}{J} \varvec{\eta } \cdot \varvec{A}: ({{\,\mathrm{\nabla ^{\varvec{x}}}\,}}{\delta \varvec{u}} \varvec{F}) \cdot \varvec{F}^T \cdot \varvec{n} \hbox {d}A \nonumber \\ =&\int _{\Gamma } \varvec{\eta } \cdot \varvec{a}: {{\,\mathrm{\nabla ^{\varvec{x}}}\,}}{\delta \varvec{u}} \cdot \varvec{n} \hbox {d}A, \end{aligned}$$

where \(\varvec{a}\) is defined in (35).

Fig. 18

Model of a bar under a uniaxial loading

Fig. 19

Stress–strain curves with different meshes for local (loc) and nonlocal damage (non) models

Verification of the local nonlocal damage model

We here adopt a uniaxial loading example to verify the nonlocal damage model. As shown in Fig. 18, a bar with a length of 10 m is fixed on the left bound and subjected to an incremental displacement on the right bound. The material is described by an isotropic elastic damage model. The elastic modulus is 10 GPa, Poisson’s ratio is 0.2, and \(\varepsilon _0\) and \(\varepsilon _f\) in (30) are \(1\times 10^{-4}\) and \(2\times 10^{-3}\), respectively. We set the size of element as \(h=0.5, 1\) and 2m. The internal length for the integral regularization is 2.2 m. Figure 19 plots the curves of \(\varepsilon -\sigma \) for different mesh sizes and constitutive models. We can see that the results are mesh-dependent for local damage model and less dissipation energy for the finer grid. On the contrary, the results are identical for different meshes with the nonlocal damage model. We further compare the distribution of strain along the horizontal axis and find that the distribution is almost uniform for the nonlocal damage model and the strain localizes at the rightmost element for the local damage model. We thus believe our implementation of the nonlocal damage model is correct.

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Liu, C., Sun, W. Shift boundary material point method: an image-to-simulation workflow for solids of complex geometries undergoing large deformation. Comp. Part. Mech. 7, 291–308 (2020).

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  • Material point method
  • Shift domain
  • Image-based simulations
  • Nonlocal damage
  • Granular materials