Convected Particle Domain Interpolation Research Articles

Generalized particle domain method: An extension of material point method generates particles from the CAD files

SummaryIn this paper, a generalized particle domain method (GPDM) is proposed and developed within the framework of the convected particle domain interpolation method. This new method generates particles directly from non‐uniform rational B‐spline (NURBS)‐based CAD file of a continuum body. The particle domain corresponds to a NURBS element even for trimmed elements of solids with complex geometries. The shape functions and the gradient of shape functions are evaluated using NURBS basis functions to map material properties between particles and grid nodes. It approves that this proposed GPDM can track the domain of particles accurately and avoid the issue of cell‐crossing instability. Several numerical examples are presented to demonstrate the high performance of this proposed new particle domain method. It is shown that the results obtained using the proposed GPDM are consistent with the experimental data reported in the literature. Further development of the generalized particle domain method can provide a link to the material point method and isogeometric analysis.

Multiple discrete crack initiation and propagation in Material Point Method

Cracks in MPM (CRAMP) is one of the most prominent discrete crack simulation methods in the Material Point Method (MPM) due to its simplicity and versatility. However, CRAMP is yet to include the capability to simulate concurrent crack initiations and propagations, as well as propagation to the edge of the material domain. The method proposed in this paper enables the simulation of multiple crack paths with CRAMP via the dynamic assignment of particles to separate grids while minimizing the number of necessary grids. It also proposes methods of evaluating crack initiation and propagation via the Rankine criterion. The proposed methods are then implemented in an in-house Convected Particle Domain Interpolation (CPDI) MPM developed at Aalto University.To verify the integrity of the CPDI algorithm, our CPDI code with the proposed method implemented simulated a CPDI vortex. Furthermore, six fracture-simulation verification test cases were carried out: (1) through-crack in an infinite plate; (2) mode-I propagation; (3) initiation; (4) initiation with large deformations; (5) merging; (6) multiple initial cracks; and (7) radially-cracked thick ring. All these verification tests show successful initiation, propagation, merging, crack opening, and agreement with the results from the literature, as well as the convergence of various parameters with the expected rates.

B-spline convected particle domain interpolation method

The B-spline material point method (BSMPM) has been shown to be a promising approach to effectively mitigate the cell-crossing instability by using high-order B-spline basis functions. However, the BSMPM still suffers from the quadrature error, another well-known source of inaccuracy in the material point methods. On the other hand, the convected particle domain interpolation (CPDI) method effectively reduces the quadrature error by using alternative basis functions. The B-spline basis functions can be naturally used within the CPDI formulation, here called the BSCPDI method. The BSCPDI may be prone to the cell-crossing instability since the alternative basis functions in the BSCPDI are always C0-continuous despite the high-continuity of the B-spline basis functions used as the original grid basis functions. In this paper, the reproducibility of the BSCPDI effective basis function gradients and the susceptibility of the method to the cell-crossing instability are investigated analytically and numerically. The effectiveness of the BSCPDI in mitigating the quadrature error is also investigated through numerical simulations.

Shell Structure Analysis Based on the Convected Particle Domain Interpolation

Shell Structure Analysis Based on the Convected Particle Domain Interpolation

An improved convected particle domain interpolation material point method for large deformation geotechnical problems

AbstractA novel material point method called the improved convected particle domain interpolation (ICPDI) is proposed in this paper to simulate large‐deformation problems in geotechnical engineering. The ICPDI framework is based on the existing convected particle domain interpolation (CPDI) and employs an adaptive orthogonal improved interpolating moving least square (AOIIMLS) shape functions. The AOIIMLS shape functions prevent computational instability problems caused by singular or ill‐conditioned matrices through avoiding the direct solving of inverse matrices. Additionally, ICPDI adopts the velocity gradient to calculate the velocity field in particle domains and reduces cell‐crossing errors by using AOIIMLS shape functions. To reconstruct the velocity function in the background grid, the velocities of the material points and four related corner points are employed through AOIIMLS shape functions. The bar axis‐aligned vibration, ring radial expansion, slope elasto‐plastic slumping, and post‐failure tunnel collapse behavior examples are used to verify the accuracy and effectiveness of ICPDI by comparing the results with other numerical methods and experiments. Numerical simulation results show that ICPDI outperforms the material point method (MPM) and CPDI in terms of computational accuracy and convergence while reducing cell‐crossing errors. Consequently, this study demonstrates the capability of ICPDI in simulations of large‐deformation geotechnical problems.

Coupling Explicit Phase-field MPM for Two-Dimensional Hydromechanical Fracture in Poro-elastoplastic Media

In this paper, a novel coupling explicit phase-field material point method with the convected particle domain interpolation (ePF-CCPDI) is proposed to predict the large deformation elastoplastic failure behavior in two-dimensional fully saturated porous media. This method provides a new thermodynamic derivation for the explicit phase-field modeling of multi-physical fracture problems. To account for the coupling effect of pore fluid flow on fractured porous solids, the pressure energy is included in the coupled system energy balance equation. The constitutive relationship of saturated porous media indicates that the pore pressure has a direct effect on the deformation of the solid skeleton. For achieving the fluid pressure smoothly distributed in damaged porous media under large deformation, exponential indicator functions are constructed to interpolate the fluid properties from the intact medium to the fully broken one. Meanwhile, the crack-opening-dependent permeability is introduced to estimate fluid flow inside the crack. The coupled explicit phase-field plasticity model is adopted to characterize the elastoplastic failure behavior of the solid skeleton. Moreover, a mixed explicit-implicit staggered solution scheme is developed to effectively solve the governing equations of the coupled system. To eliminate the numerical noises while material points cross the cell boundaries in large deformation simulation, the improved convected particle domain interpolation technique (CPDI) is adopted to enhance the computational accuracy. Two solution schemes are also proposed based on the extended material point penalty method to handle the specified pressure boundary condition. Finally, several representative two-dimensional numerical examples are conducted to validate the accuracy and capability of the proposed method.

A coupled model of asymmetric GIMP and tetrahedron CPDI based on the penalty contact algorithm for simulating dynamic rock splitting

A coupled material point model (MPM) based on the asymmetric generalized interpolation material point (aGIMP) and the tetrahedron convected particle domain interpolation (CPDI-Tet4) was established to simulate the dynamic splitting of a rock Brazilian disc under small deformation. Furthermore, a boundary-based contact algorithm was developed to accurately describe the contact force between the split bar's aGIMP particles and the rock's CPDI-Tet4 particles. The proposed aGIMP technology enabled particles to move on a non-uniform background grid by changing the original basis functions. A good simulation effect was achieved. It was concluded that the rock particle size plays an important role in the accuracy of the contact algorithm. Two shear failure bands were generated near the contact zones first after the rock specimen reached the force equilibrium state, and then the four arc fracture surfaces grew from the two sides of the shear failure bands and coalesced to a radial fracture surface on the mid-cross section of the specimen. In addition, the tensile strength calculated by the MPM model exhibited the typical rate effect of rock material and was slightly lower than the practical tensile strength of the particle on the radial fracture surface.

Explicit phase-field material point method with the convected particle domain interpolation for impact/contact fracture in elastoplastic geomaterials

A novel explicit phase-field material point method with the convected particle domain interpolation (ePF-CPDI) is proposed for solving large deformation dynamic impact/contact fracture problems in elastoplastic geomaterials. In this method, we derive an explicit rate-dependent phase-field fracture model relying on the microforce balance law and the second law of thermodynamics. A coupled explicit phase-field plasticity model is then developed to describe dynamic elastoplastic fracture responses of geomaterials. Here the explicit time integration strategy and the staggered solution scheme are utilized in this work to solve the coupled-field governing equations based on the material point method. To eliminate the numerical noises caused by material points crossing cell boundaries, the convected particle domain interpolation technique is adopted to improve the computational accuracy in large deformation simulation. Furthermore, the proposed approach combining with the particle to particle contact algorithm is extended to deal with more complicated high-velocity impact and multi-body contact elastoplastic fracture problems. Numerical experiments are employed to validate the high accuracy and excellent capability of the proposed method and discuss the influences of main parameters on the phase-field fracture modeling.

Coupled GIMP and CPDI material point method in modelling blast-induced three-dimensional rock fracture

Three-dimensional rock fracture induced by blasting is a highly complex problem and has received considerable attention in geotechnical engineering. The material point method is firstly applied to treat this challenging task. Some inherent weaknesses can be overcome by coupling the generalized interpolation material point (GIMP) and the convected particle domain interpolation technique (CPDI). For the media in the borehole, unchanged GIMP-type particles are used to guarantee a homogenous blast pressure. CPDI-Tetrahedron type particles are employed to avoid the fake numerical fracture near the borehole for the rock material. A blasting experiment using three-dimensional single-borehole rock was simulated to examine the applicability of the coupled model under realistic loading and boundary conditions. A good agreement was achieved between the simulation and experimental results. Moreover, the mechanism of three-dimensional rock fracture was analyzed. It was concluded that rock particle size and material parameters play an important role in rock damage. The reflected tensile waves cause severe damage in the lower part of the model. Rayleigh waves occur on the top face of the rock model to induce a hoop failure band.

An adaptive interpolation material point method and its application on large-deformation geotechnical problems

A novel Material Point Method (MPM) called the Adaptive Interpolation Material Point Method (AIMPM) is proposed to simulate large deformation problems in this work. The proposed method combines the domain interpolation proposed by the Convected Particle Domain Interpolation (CPDI) with the particle interpolation from the original MPM to address large deformation problems of geotechnical materials. The novelty of the model is twofold. First, the 3D domain interpolation formula is extended, and the volume term in the shape function is calculated from the corner points characterizing the material configuration. Second, the particle interpolation and the domain interpolation are coupled to accurately reflect the stress-strain behavior of geotechnical materials under large deformation. The two interpolation methods can be transformed adaptively through the criterion of Jacobian determinant or equivalent plastic strain. The performance of the AIMPM is verified by axial vibration, generalized vortex, rectangular collision, and column collapse cases. The simulation results show that the AIMPM significantly reduces the cell crossing error by domain interpolation compared to the MPM. More importantly, AIMPM is more accurate in capturing the exact location of geotechnical material flow. The work described in this study demonstrates the capability of the AIMPM in simulations of large-deformation geotechnical problems.

Phase-field implicit material point method with the convected particle domain interpolation for brittle–ductile failure transition in geomaterials involving finite deformation

A phase-field implicit material point method with the convected particle domain interpolation (PF-ICPDI) is proposed to model the brittle–ductile failure transition in pressure-sensitive geomaterials involving finite deformation. In this method, the phase-field fracture formulation relying on the microforce balance law and the second law of thermodynamics is adopted as a nonlocal damage function for the elastoplastic fracture analysis. A smooth two-yield-surface plasticity constitutive model is utilized to evaluate the brittle–ductile failure transition behavior of pressure-sensitive geomaterials. The coupling effect of phase-field fracture model and cap plasticity model is established by introducing an effective phase-field stress and splitting the total stored energy into elastic and plastic parts. The implicit material point method that can avoid severe mesh distortion and improve numerical stability is then developed to solve the quasi-static elastoplastic fracture finite deformation problem. Furthermore, the convected particle domain interpolation technique is adopted to eliminate numerical noises and improve computational accuracy while material points crossing cell boundaries in modeling the large deformation brittle–ductile failure transition process. The staggered incremental iterative scheme is carried out to solve the coupled discrete governing equations. The accuracy and capability of the proposed PF-ICPDI method are demonstrated through several representative numerical examples, as well as the effects of main material parameters on the phase-field brittle–ductile failure transition modeling of geomaterials are discussed.

Multi-surfaced elasto-plastic wood material model in material point method

Wood is a naturally occurring material widely used for construction. Due to its natural origin, wood properties vary and its behaviour is complex. This paper shows an implementation of a multi-surface elasto-plastic constitutive material model for wood into a custom explicit material point method code. The constitutive model chosen is one proposed by Schmidt & Kaliske with minor modifications to ensure better internal consistency. The model parameters are chosen based on literature data for spruce. The paper presents two Convected Particle Domain Interpolation Material Point Method simulations of experiments, both performed with the previously established model parameters. The first simulation replicates a compression test of a spruce specimen perpendicular to grain direction, carried out at the Department of Civil Engineering, Aalto University. The second simulation replicates an experiment from literature, in which a spruce specimen with knots is tensioned until failure. The numerical simulations successfully replicate the experimental outcomes qualitatively in terms of the deformation and load-displacement curves. Simulations of the three knotted specimens under tension, with introduced slight variation in wood grain direction, replicate different failure patterns with a similar failure load, resembling the behaviour of natural wooden structural elements. Additionally, one of the obtained failure patterns replicates that of the experiment well.

From solid to disconnected state and back: Continuum modelling of granular flows using material point method

This paper introduces Granular algorithm enhancing Material Point Method (MPM) that allows for simulation of granular flows in the quasi-static state, the moderate flow state and the disconnected flow state. The paper first shows the shortcomings of MPM algorithms in modelling the different states of granular flows. Next, it proposes Granular MPM, an enhancement that explicitly introduces the different states of granular flow into MPM and defines the rules for the transition between those states. Subsequently, the paper gives the algorithm and implementation for Granular MPM. The provided algorithm can enhance the common versions of MPM, including original MPM, Generalised Interpolation Material Point and Convected Particle Domain Interpolation. Finally, the paper evaluates Granular MPM and compares it with other available formulation based on: (i) an examination of the behaviour of granular points on a slope, (ii) a simulation of a granular flow over two disconnected inclined surfaces, (iii) a simulation of a silo filling and (iv) a simulation of Toyoura sand flow experiment. The results of Granular MPM simulations are significantly more realistic when compared to the results obtained by other available MPM formulations. The results also indicate that Granular MPM allows for more accurate replication of steady state flows and reduces the grid dependency of MPM when modelling the disconnected flow state, as the initial contact is independent from the grid size.

Implicit Material Point Method with Convected Particle Domain Interpolation for Consolidation and Dynamic Analysis of Saturated Porous Media with Massive Deformation

A coupling convected particle domain interpolation based implicit material point method (ICCPDI) is developed in this paper for the long-time consolidation and dynamic problems of fully saturated porous media involving massive deformation. In the method, the implicit formulas based on the u-p form governing equations are derived to overcome the time step size limitation within the conventional explicit method and in turn improve the computational efficiency particularly for consolidation problems. Then, the CPDI technique is adopted to eliminate the numerical noises occurring while material points crossing cell boundaries in the massive deformation of solid skeleton. Moreover, an extended material point penalty method is proposed to effectively treat the boundary constraints in the coupled system by avoiding introducing an auxiliary boundary layer. Numerical simulation results of several representative examples demonstrate the accuracy and efficiency of the proposed ICCPDI method to predict the consolidation and dynamic responses of saturated porous media with massive deformation.

Comparison of UBCSAND and Hypoplastic soil model predictions using the Material Point Method

Liquefaction effects predicted by implementing two well-known constitutive models for sandy soils are compared; i.e., the Hypoplastic soil model (HPS) and the UBCSAND model. To numerically simulate dynamic loading, the Convected Particle Domain Interpolation (CPDI) method, an advanced Material Point Method (MPM), is employed within a multi-phase framework. The numerical results from the UBCSAND model are compared against published experimental data. A comparison between the performance of UBCSAND and HPS model is presented by calibrating the former’s constitutive behaviour for Berlin sand to that of the HPS model. The shake table test performed at Rensselaer Polytechnic Institute is numerically simulated and results are compared against the published experimental data. Good agreement between the experiment and the numerical calculation is obtained. Thereafter, models are employed to numerically simulate a driven pile installation. The results from both models are compared against published experimental data. The multi-phase CPDI formulation is shown to be capable of reproducing liquefaction for the the pile driving example.

A Total-Lagrangian Material Point Method for solid mechanics problems involving large deformations

The material point method (MPM) has found successful applications in many engineering problems involving large displacement, large deformation and contacts. The standard MPM formulation, which adopts piece-wise linear basis functions, suffers from the so-called cell-crossing instability, low order of convergence and numerical fracture. Modifications have been made to this standard MPM to mitigate these issues: B-spline MPM (BSMPM), the generalized interpolation material point (GIMP) and convected particle domain interpolation (CPDI) all decrease cell-crossing instabilities and increase the order of convergence, but only CPDI effectively suppresses numerical fracture. However, these methods, CPDI in particular, significantly increase the method’s implementation and computational complexity. This paper presents a total Lagrangian MPM, dubbed TLMPM, that overcomes the issues of the conventional MPM while being more efficient and easier to implement than CPDI. The method is used for impact analyses of a cylinder bar made of steel and necking and fracture of cylinder alloy specimens. The numerical solutions are in satisfactory agreement with the experiment data. No numerical fracture occurred for simulations involving very large tensile deformation without special treatment such as done in the CPDI. Convergence analyses using the method of manufactured solutions show that the TLMPM is second-order accurate for problems of which boundary is axis-aligned. For the challenging generalized vortex problem, it also converges quadratically for relatively coarse meshes. Moreover, the model is able to simulate physically based fracture using continuum damage mechanics.

A convected particle least square interpolation material point method

SummaryApplying the convected particle domain interpolation (CPDI) to the material point method has many advantages over the original material point method, including significantly improved accuracy. However, in the large deformation regime, the CPDI still may not retain the expected convergence rate. The paper proposes an enhanced CPDI formulation based on least square reconstruction technique. The convected particle least square interpolation (CPLS) material point method assumes the velocity field inside the material point domain as nonconstant. This velocity field in the material point domain is mapped to the background grid nodes with a moving least squares reconstruction. In this paper, we apply the improved moving least squares method to avoid the instability of the conventional moving least squares method due to a singular matrix. The proposed algorithm can improve convergence rate, as illustrated by numerical examples using the method of manufactured solutions.

On the use of domain-based material point methods for problems involving large distortion

Challenging solid mechanics problems exist in areas such as geotechnical and biomedical engineering which require numerical methods that can cope with very large deformations, both stretches and torsion. One candidate for these problems is the Material Point Method (MPM), and to deal with stability issues the standard form of the MPM has been developed into new domain-based techniques which change how information is mapped between the computational mesh and the material points. The latest of these developments are the Convected Particle Domain Interpolation (CPDI) approaches. When these are demonstrated, they are typically tested on problems involving large stretch but little torsion and if these MPMs are to be useful for the challenging problems mentioned above, it is important that their capabilities and shortcomings are clear. Here we present a study of the behaviour of some of these MPMs for modelling problems involving large elasto-plastic deformation including distortion. This is carried out in a unified implicit quasi-static computational framework and finds that domain distortion with the CPDI2 approaches affects some solutions and there is a particular issue with one approach. The older CPDI1 approach and the standard MPM however produce physically realistic results. The primary aim of this paper is to raise awareness of the capabilities or otherwise of these domain-based MPMs.

A convected-particle tetrahedron interpolation technique in the material-point method for the mesoscale modeling of ceramics

Convected particle domain interpolation, which is known to boost the accuracy of the material-point method, is applied in a form called convected-particle tetrahedron interpolation (CPTI). CPTI exploits the efficiency of tetrahedral tessellations to represent complex structural geometries, while still solving field equations on a rectilinear background grid. Advantages include anti-locking and an ability to handle extremely large deformations without suffering typical Eulerian advection errors. CPTI is demonstrated to resolve long-standing errors caused by spuriously ragged (stair-stepped) surfaces, and it is also shown to accommodate mathematically rigorous evaluation of surface integrals in models for contact and friction. Benefits of this work are illustrated in mesoscale simulations of an aluminum oxynitride ceramic.

Enhancement of the material point method using B‐spline basis functions

The material point method (MPM) enhanced with B‐spline basis functions, referred to as B‐spline MPM (BSMPM), is developed and demonstrated using representative quasi‐static and dynamic example problems. Smooth B‐spline basis functions could significantly reduce the cell‐crossing error as known for the original MPM. A Gauss quadrature scheme is designed and shown to be able to diminish the quadrature error in the BSMPM analysis of large‐deformation problems for the improved accuracy and convergence, especially with the quadratic B‐splines. Moreover, the increase in the order of the B‐spline basis function is also found to be an effective way to reduce the quadrature error and to improve accuracy and convergence. For plate impact examples, it is demonstrated that the BSMPM outperforms the generalized interpolation material point (GIMP) and convected particle domain interpolation (CPDI) methods in term of the accuracy of representing stress waves. Thus, the BSMPM could become a promising alternative to the MPM, GIMP, and CPDI in solving certain types of transient problems.