Extended B-spline-based mixed material point method stabilized by the variational multiscale method for compressible and nearly incompressible hyperelastic materials

Riichi Sugai; Reika Nomura; Shuji Moriguchi; Kenjiro Terada

doi:10.3934/acse.2025013

Article Contents

June 2025, Volume 4, 85-118. Doi: 10.3934/acse.2025013

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Extended B-spline-based mixed material point method stabilized by the variational multiscale method for compressible and nearly incompressible hyperelastic materials

1.
Department of Civil and Environmental Engineering, Tohoku University, 6-6-06 Aza-Aoba, Aramaki, Aoba-ku, Sendai, 980-8579, Miyagi, Japan
2.
International Research Institute of Disaster Science, Tohoku University, 468-1 Aza-Aoba, Aramaki, Aoba-ku, Sendai, 980-8572, Miyagi, Japan

^*Corresponding author: Kenjiro Terada
^*Corresponding author: Kenjiro Terada

Received: January 2025

Revised: April 2025

Published online: April 15, 2025

The first author is supported by JST SPRING, Grant Number JPMJSP2114.

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Abstract

To further improve the stability of the extended B-spline (EBS) based material point method (EBSMPM), a displacement-pressure ($ \mathit{\boldsymbol{u}} $-$ p $)-mixed formulation incorporating the variational multiscale (VMS) method is formulated within the finite strain framework. In this formulation, we define a constraint on the pressure field that can be applied to both compressible and nearly incompressible materials, taking into account the extensibility to material models that exhibit plastic incompressibility. In order to satisfy the LBB conditions for ensuring stability in the mixed formulation with equal-order interpolation, the VMS method is formulated within finite strain framework in computational solid mechanics. The adoption of the VMS method makes it possible to use higher-order EBS basis functions of the same order for interpolation of both displacement and pressure, and this successfully prevents the occurrence of numerical instability. Three representative numerical examples are presented to demonstrate the performance of the proposed method for solid mechanics problems using compressible and nearly incompressible materials, comparing the formulation whose independent variable is only the displacement field. Additionally, through these numerical examples, the ability to suppress volumetric locking and pressure oscillation is also verified.

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Mathematics Subject Classification: Primary: 74G15, 65N85; Secondary: 74B20.

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Figure 1. Schematic view of the MPM

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Figure 2. Classification of quadratic B-spline basis functions and associated control points (the upper panel), and composition of the EBS basis functions in the $ i $th direction

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Figure 3. Geometry and boundary conditions for Example 1

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Figure 4. Convergence trend of vertical displacements for compressible material obtained for Example 1. (A), (B), and (C) are obtained using the irreducible formulation, $ \mathit{\boldsymbol{u}} $-$ p $ mixed formulation without VMS, and $ \mathit{\boldsymbol{u}} $-$ p $ mixed formulation with VMS, respectively

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Figure 5. Pressure distribution for compressible material in Example 1

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Figure 6. Convergence trend of vertical displacements for nearly incompressible material obtained for Example 1. (A) and (B) are obtained using the irreducible formulation and $ \mathit{\boldsymbol{u}} $-$ p $ mixed formulation with VMS, respectively

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Figure 7. Pressure distribution for nearly incompressible material in Example 1

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Figure 8. Geometries and boundary conditions for Example 2

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Figure 9. Convergence trend of the reaction forces for compressible material obtained for Example 2. (A), (B), and (C) are obtained using the irreducible formulation, $ \mathit{\boldsymbol{u}} $-$ p $ mixed formulation without VMS, and $ \mathit{\boldsymbol{u}} $-$ p $ mixed formulation with VMS, respectively

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Figure 10. Pressure distribution for compressible material in Example 2

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Figure 11. Convergence trend of the reaction forces for nearly incompressible material obtained for Example 2. (A), (B), and (C) are obtained using the irreducible formulation, $ \mathit{\boldsymbol{u}} $-$ p $ mixed formulation without VMS, and $ \mathit{\boldsymbol{u}} $-$ p $ mixed formulation with VMS, respectively

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Figure 12. Pressure distribution for nearly incompressible material in Example 2

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Figure 13. Analysis conditions for Example 3. (A) shows geometry with boundary conditions, and (B) illustrates configuration of the physical domain with the background grid cells

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Figure 14. Strain energy for compressible material obtained for Example 3. (A), (B), and (C) are obtained in the irreducible formulation, $ \mathit{\boldsymbol{u}} $-$ p $ mixed formulation without VMS, and $ \mathit{\boldsymbol{u}} $-$ p $ mixed formulation with VMS, respectively

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Figure 15. Pressure distributions for compressible material in Example 3

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Figure 16. Strain energy obtained using the $ \mathit{\boldsymbol{u}} $-$ p $ mixed formulation with VMS in the nearly incompressible case in Example 3

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Figure 17. Pressure distribution for nearly incompressible material in Example 3

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Figure 18. Parameter study for the algorithmic parameters, $ c_{1} $ and $ c_{2} $, with the geometry and the boundary conditions given in Example 1

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Table 1. Reference abbreviations for simulation cases employing $ d $th order basis function

Abbreviation	Method	Primal variable	Stabilization
MPM-$ Q_{d} $	EBSMPM	$ \mathit{\boldsymbol{u}} $	-
MPMup-$ Q_{d}/Q_{d} $	EBSMPM	$ \mathit{\boldsymbol{u}} $ and $ p $	-
MPMupVms-$ Q_{d}/Q_{d} $	EBSMPM	$ \mathit{\boldsymbol{u}} $ and $ p $	VMS
FEM-$ Q_{d} $	Generalized FEM-like	$ \mathit{\boldsymbol{u}} $	-
FEMup-$ Q_{d}/Q_{d} $	Generalized FEM-like	$ \mathit{\boldsymbol{u}} $ and $ p $	-
FEMupVms-$ Q_{d}/Q_{d} $	Generalized FEM-like	$ \mathit{\boldsymbol{u}} $ and $ p $	VMS

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Table 2. Common parameters in numerical examples

Parameter		Value		Unit
		Compressible	Nearly incompressible
Bulk modulus	$ \kappa $	$ 200.471 $	$ 400,942 $	MPa
Shear modulus	$ \mu $	$ 92.525 $	$ 80.194 $	MPa
Algorithmic parameters	$ c_{1} $	$ 0.1 $		-
	$ c_{2} $	$ 1.0 $		-
Occupation threshold	$ \phi_{\rm cr} $	$ 0.01 $		-
Penalty parameter	$ \beta $	$ 1.0\times10^{5} $		N/mm

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