Difference between revisions of "MPM Methods and Simulation Timing"
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Two versions of CPDI have appeared. The first version finds the edges of the deformed domain using the deformation gradient on the particle. In this approach, an initial 2D rectangle deforms to a parallelogram and an initial 3D orthogonal box deforms to a parallelepiped.<ref name='CPDI'/> The coordinates for the corners of the deformed domain and this a function of the initial domain shape and the particle's deformation gradient. The final shape functions depends on the finite element shape functions used for the particle domain. For example lCPDI using linear shape functions (4 nodes for 2D domains and 8 nodes for 3D domains) while qCPDI using quadratic shape function (8 nodes for 2D domains). The options for "other..." versions allow for alternative finite element discretizations of the particle domains. | Two versions of CPDI have appeared. The first version finds the edges of the deformed domain using the deformation gradient on the particle. In this approach, an initial 2D rectangle deforms to a parallelogram and an initial 3D orthogonal box deforms to a parallelepiped.<ref name='CPDI'/> The coordinates for the corners of the deformed domain and this a function of the initial domain shape and the particle's deformation gradient. The final shape functions depends on the finite element shape functions used for the particle domain. For example lCPDI using linear shape functions (4 nodes for 2D domains and 8 nodes for 3D domains) while qCPDI using quadratic shape function (8 nodes for 2D domains). The options for "other..." versions allow for alternative finite element discretizations of the particle domains. | ||
In CPDI2, the corner locations are separately tracked during the analysis and their current locations are used to define the deformed particle domain. | In CPDI2, the corner locations are separately tracked during the analysis and their current locations are used to define the deformed particle domain. For example, a 2D domain deforms to an arbitrary quadrilateral and a 2D domain an arbitrary 8-cornered domain. | ||
== Theory:Stress and Strain Updates == | == Theory:Stress and Strain Updates == | ||
Revision as of 22:24, 5 September 2013
These command select the MPM method to use and control time step and total time for the simulation.
Theory: Shape Functions
Many tasks in MPM involve extrapolations from particles to the grid or from the grid to the particles. These extrapolations are controlled by "Shape Functions," and the various MPM methods correspond to various methods for evaluating the shape functions. In the most generalized description of MPM (called GIMP for General Interpolation Material Point[1]), the shape function (Sip) and shape function gradient (Gip) for the node i/particle p pair are:
[math]\displaystyle{ S_{ip} = {\int_{\Omega_p} \chi_p(\vec x)N_i(\vec x) dV\over \int_{\Omega_p} \chi_p(\vec x) dV}\qquad{\rm and}\qquad G_{ip} = {\int_{\Omega_p} \chi_p(\vec x)\nabla N_i(\vec x) dV\over \int_{\Omega_p} \chi_p(\vec x) dV} }[/math]
where Ωp is the domain for particle p, χp(x) is the particle basis shape function for particle p, and Ni(x) is grid shape function for node i. The various MPM methods depend on the choice of χp(x) and on the method for evaluating the shape function integrals. The chart on the right shows an MPM "genealogy" or shows how the various MPM methods descend from GIMP.
Classic MPM
The first decision is to choose χp(x). If this function is set to the Dirac delta function (δ(x)), the shape function reduced to the grid shape function, which recovers "Classic MPM" or the original derivation of MPM.[2] This original MPM method is not very robust for large displacement problems because particles crossing cell boundaries in the mesh can cause significant noise.
uGIMP
MPM results are greatly improved by choosing χp(x) to be a Heaviside function or to be 1 in the particle domain and zero elsewhere. All current MPM styles descend from this selection. In principle, one could chose other χp(x) (e.g., a Gaussian function centered on the particle) and derive new MPM options, but this area is unexplored. The use of the Heaviside function leads to simplification of both Sip and Gip, but the integrals still need to be evaluated.
The simplest approach is to assume the particle domain has the same shape as the initial domain and merely translates with particle motion. This method, which is called uGIMP (for undeformed GIMP), has the greatest efficiency because shape functions can be explicitly calculated for all possible particle locations. uGIMP works well for small and modest deformation. Very large tension strains (>50%) can cause particles to numerically separate because the undeformed domains no longer interact with each other.
Finite GIMP
In finite GIMP, the integral is over the deformed particle domain. Although this has been shown to give improved results in 1D (where it is practical[1]), it is too inefficient for arbitarary deformation of 2D and 3D particle domains. As a result, finite GIMP is not used.
CPDI
A useful improvement for large deformation is CPDI or the Convected Partical Domain Integration approach.[3] In CPDI, the integral over the deformed particle domain is make practical by approximating the integrand. In brief:
- The deformed particle domain is discretized by isoparametric finite elements. In practice, the discretization uses a single element, but it could use more.
- The Integrand for Sip and Gip are then approximate by expanding them in the finite element shape functions
- Integration over then shape functions gives the final results.
Two versions of CPDI have appeared. The first version finds the edges of the deformed domain using the deformation gradient on the particle. In this approach, an initial 2D rectangle deforms to a parallelogram and an initial 3D orthogonal box deforms to a parallelepiped.[3] The coordinates for the corners of the deformed domain and this a function of the initial domain shape and the particle's deformation gradient. The final shape functions depends on the finite element shape functions used for the particle domain. For example lCPDI using linear shape functions (4 nodes for 2D domains and 8 nodes for 3D domains) while qCPDI using quadratic shape function (8 nodes for 2D domains). The options for "other..." versions allow for alternative finite element discretizations of the particle domains.
In CPDI2, the corner locations are separately tracked during the analysis and their current locations are used to define the deformed particle domain. For example, a 2D domain deforms to an arbitrary quadrilateral and a 2D domain an arbitrary 8-cornered domain.
Theory:Stress and Strain Updates
Commands
References
- ↑ 1.0 1.1 S. G. Bardenhagen, J. E. Guilkey, K. M. Roessig, J. U. Brackbill, W. M. Witzel, and J. C. Foster, "An Improved Contact Algorithm for the Material Point Method and Application to Stress Propagation in Granular Material," Computer Modeling in Engineering & Sciences, 2, 509-522 (2001).
- ↑ D. Sulsky, Z. Chen, and H. L. Schreyer, "A Particle Method for History-Dependent Materials,&wuot; Comput. Methods Appl. Mech. Engrg., 118, 179-186 (1994).
- ↑ 3.0 3.1 A. Sadeghirad, R. M. Brannon, and J. Burghardt, "A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations," Int. J. Numer. Meth. Engng (2011)