# An investigation of stress inaccuracies and proposed solution in the material point method

## Abstract

Stress inaccuracies (oscillations) are one of the main problems in the material point method (MPM), especially when advanced constitutive models are used. The origins of such oscillations are a combination of poor force and stiffness integration, stress recovery inaccuracies, and cell crossing problems. These are caused mainly by the use of shape function gradients and the use of material points for integration in MPM. The most common techniques developed to reduce stress oscillations consider adapting the shape function gradients so that they are continuous at the nodes. These techniques improve MPM, but problems remain, particularly in two and three dimensional cases. In this paper, the stress inaccuracies are investigated in detail, with particular reference to an implicit time integration scheme. Three modifications to MPM are implemented, and together these are able to remove almost all of the observed oscillations.

## Keywords

Double mapping Material point method Shape functions Stress oscillation## Abbreviations

- CDPI
Convected domain particle interpolation

- CMPM
Composite material point method

- DDMP
Dual domain material point

- DM
Double mapping

- DM-C
Double mapping using CMPM

- DM-G
Double mapping using GIMP shape functions

- DM-GC
Double mapping using GIMP shape functions and CMPM

- FEM
Finite element method

- FE
Finite element

- GIMP
Generalized interpolation material point

- GM
Gauss mapping

- MP
Material point

- MPM
Material point method

- SD
Material point support domain

- SF
Shape function

## Latin symbols

**a**Acceleration vector

**a**_{p}Material point acceleration

- \( \overline{\mathbf{a}} \)
Vector of nodal accelerations

- A
Constant derived from the axisymmetric solution

- A
_{1} Constant derived from the axisymmetric boundary conditions

**b**Body forces

**B**Strain-displacement matrix

**B**^{ax}Strain-displacement matrix for axisymmetric domain

**B**^{C}Strain-displacement matrix for CMPM patch

- c
_{p} Soil peak cohesion

- c
_{r} Soil residual cohesion

- C
Constant derived from the axisymmetric solution

- cmp
Current number of material points in the element

- d
Distance between the element boundary and the axisymmetric internal boundary

**D**_{g}Elastic matrix at the Gauss point

**D**_{i}Elastic matrix at node i

**D**_{p}Elastic matrix at the sampling point

- \( {\mathbf{D}}_{\rm p}^{\rm ax} \)
Elastic matrix at the sampling point for axisymmetric domain

- E
Young’s modulus

- elmp
Material points affecting an element

**F**^{ext}Vector of external nodal forces

**F**^{int}Vector of internal nodal forces

- F
_{mag}^{int} Internal nodal force magnitude

- F
_{VM} Von Mises yield function

- F
_{x}^{int} Nodal internal force in the horizontal direction

- F
_{y}^{int} Nodal internal force in the vertical direction

- F
_{yield} Yield function

**g**Gravity vector

- H
Height of the vertical cut benchmark

- \( \overline{{\mathbf{H}}} \)
Matrix of shape functions, representing either

**N**or matrix of S_{ip*}- H
_{s} Softening modulus

- i
Subscript representing nodal values

**J**Jacobian matrix

**J**^{mp}Jacobian matrix computed using material point shape function derivatives

- \( \left| {\mathbf{J}} \right| \)
Jacobian matrix determinant

- k
Iteration number

**K**Stiffness matrix

**K**_{el}Element stiffness matrix

- K
_{mag} Stiffness matrix magnitude

- K
_{x} Diagonal entry of the stiffness matrix corresponding to the horizontal degree of freedom

- K
_{y} Diagonal entry of the stiffness matrix corresponding to the vertical degree of freedom

**K**_{v}Global stiffness matrix

- lp
Half of the material point support domain length

- L
Length of the vertical cut benchmark

**M**Mass matrix

- m
_{p} Material point mass

- M
_{x} Diagonal entry of the mass matrix corresponding to the horizontal degree of freedom

- M
_{y} Diagonal entry of the mass matrix corresponding to the vertical degree of freedom

- N
Shape function

**N**Matrix of shape functions

- N
_{i} Nodal shape function

- ngauss
Number of Gauss points in the element

- nmp
Number of material points inside an element

- nn
Number of nodes

- \( {\mathbf{N}}_{\text{global}}^{\text{C}} \)
Matrix of global CMPM shape functions

- \( {\mathbf{N}}_{\text{local}}^{\text{C}} \)
Matrix of local CMPM shape functions

- N
^{i} CMPM shape functions, where i is the C-continuity

**N**^{i}Matrix of CMPM shape functions, where i is the C-continuity

- omp
Original number of material points in the element

- p
_{s} Applied pressure on the boundary of the axisymmetric benchmark

- r
Distance between the cylinder axis and any point inside the cylinder wall

- r
_{e} Outer boundary of the axisymmetric benchmark

- r
_{i} Internal boundary of the axisymmetric benchmark

- r
_{mp1} Radial position of a material point at the boundary of the axisymmetric benchmark

- r
_{p} Radial position of a material point

- s
Internal boundary of the axisymmetric benchmark

- smp
Number of material points with a support domain inside an element

- S
_{ip} GIMP shape functions

- S
_{ip*} Local GIMP shape functions

- t
Superscript denoting value at current time step

- t+Δt
Superscript denoting value at next time step

- u
Virtual displacement

- \( \overline{{\mathbf{u}}} \)
Vector of nodal displacements

- \(\overline{{\mathbf{u}}}^{\rm ext} \)
Vector of nodal displacements in the extended domain using CMPM

**v**Velocity vector

- \( \overline{{\mathbf{v}}} \)
Vector of nodal velocities

- V
Body volume

- V
_{p} Material point volume

**v**_{p}Material point velocity

- W
Material point integration weight

- W
^{FE} Weight associated with the Gauss point

- W*
Modified material point integration weight

- \( \overline{\text{W}} \)
Material point weight, representing either W or W*

**x**^{C}Nodal coordinates of the CMPM patch

**x**_{g}Gauss position

**x**_{p}Material point position

## Greek symbols

- α
Newmark time stepping parameter

- χ
_{p} Characteristic function

- δ
Newmark time stepping parameter

- \( \delta \overline{{\mathbf{u}}} \)
Incremental displacement

- δ
_{mp1} Material point domain

- Δq
Incremental deviatoric stress

- Δp
_{s} Incremental applied pressure on the boundary of the axisymmetric benchmark

- Δr
Mesh radial dimension for the axisymmetric domain

- \( \Delta {\varvec{\upsigma}}_{\rm p} \)
Stress increment vector at the material point

- Δσ
_{m} Incremental mean stress

- Δt
Time step

- \( \Delta \overline{{\mathbf{u}}} \)
Vector of incremental nodal displacements

- Δy
Mesh vertical dimension

- Δx
Mesh horizontal dimension

- Γ
Body surface

- η
Vertical position in local coordinates

- ν
Poisson’s ratio

- ρ
Density

- ρ
_{p} Material point density

**σ**Cauchy stress tensor

- σ
_{A} Analytical radial stress

**σ**^{ax}Cauchy stress tensor for axisymmetric domain

- σ
_{L} Stress inside a linear axisymmetric element

- σ
_{Q} Stress inside a quadratic axisymmetric element

- σ
_{θ} Tangential stress

- σ
_{r} Radial stress

- σ
_{y} Vertical stress

- σ
_{xy} Shear stress

**τ**Traction at the surface

**τ**_{p}Material point traction force

- Ω
Simulation domain

- Ω
_{p} Material point support domain

- ξ
Horizontal position in local coordinates

## 1 Introduction

GIMP [6], which distributes the influence of each material point over a characteristic or support domain, possibly extending the influence to multiple elements at a time. Both the SF and the SF gradients are modified.

CPDI [24], which is an extension of GIMP in which the material point support domain can deform, maintaining the interaction between particles even after large extension. There are a number of CPDI variants, with different orientations and behaviour of the support domain.

B-spline MPM [26], which replaces the linear SFs by functions with higher-order B-spline basis functions, which are at least C1-continuous and positive definite.

DDMP [34], which preserves the linear SFs and replaces the SF gradients by smooth continuous functions, thereby allowing the usage of a local integration procedure rather than having a material point support domain.

These techniques have been proven to reduce the impact of cell crossing. Meanwhile, other techniques use material point integration together with Gauss point integration to reduce numerical inaccuracies [2, 16]. However, a complete investigation of the causes of the stress inaccuracies has not been presented. Moreover, these techniques typically involve explicit MPM schemes, thereby ignoring the errors the proposed solutions can cause in the integration of the stiffness matrix in implicit schemes and not exploiting the advantages of implicit time integration. Finally, examples have often been investigated only for 1D cases (usually with 2D elements), so that oscillations caused by other deformations, e.g. material rotation or distortion, have not been examined.

This paper first presents the theoretical background of implicit MPM. Then, two benchmark problems are introduced to illustrate the oscillation problem. In Sect. 4, the main causes of stress oscillations are investigated. Then, a series of existing and novel solutions are presented and investigated. Finally, a comparison of regular MPM and the new proposed oscillation-free MPM is given for the simulation of a vertical cut failure, in order to demonstrate the relative performance for a problem involving both 2D geometry and elasto-plasticity.

## 2 Theoretical formulation

**σ**is the symmetric Cauchy stress tensor,

**a**is the acceleration,

**b**are the body forces, and ρ is the mass density. In MPM, because of the partition of unity of the SFs, mass is automatically conserved. The weak form of Eq. 1 including traction as a boundary condition is

**τ**is the traction at the surface Γ (i.e. the boundary condition), and V is the body volume. Following standard FEM discretisation, Eq. 2 can be expressed in the matrix form [8, 32]

**M**is the mass matrix, \( \overline{\textbf{a}} \) is the vector of nodal accelerations,

**K**is the stiffness matrix, \( \overline{\textbf{u}} \) is the vector of nodal displacements, and

**F**

^{ext}and

**F**

^{int}are the external and internal force vectors, respectively. A quasi-static formulation is obtained by removing the \( {\textbf{M}}\overline{\textbf{a}} \) term from Eq. 3. Using the Gauss–Legendre quadrature rule and discretising the continuum body using a finite set of material points, the element (nodal) mass matrix can be expressed as

_{ρ}is the material point density,

**N**is the matrix of SFs evaluated at the material point position

**x**

_{p},

**J**is the Jacobian matrix, and W is the material point integration weight (which is dimensionless and equal to the volume of the material point in local coordinates).

**K**can be expressed in terms of a small or large strain formulation, but for simplicity it is expressed here in the small strain formulation (for details of the large strain formulation, see [32]), as

**B**is the strain–displacement matrix and

**D**

_{p}is the elastic matrix at the sampling point. The element (nodal) external forces

**F**

^{ext}considering gravity and boundary tractions are

**g**is the gravity vector. The element (nodal) internal forces

**F**

^{int}are

**σ**

_{p}is the vector of material point stresses. Details of the axisymmetric form of the previous equations are presented in “Appendix A1”. Using Newmark’s [20] time integration scheme,

For an elasto-plastic material, stresses which are found to exceed the yield surface are redistributed using a consistent plastic return algorithm such that a new body force is calculated, and Eq. 10 is again solved to give plastic deformations. This is iteratively performed until no stresses exceed the yield surface. For more details see, for example, Bathe [8].

### 2.1 Material point method

## 3 Benchmarks

Two benchmarks are introduced to demonstrate and investigate the inaccuracies which occur in MPM. The first benchmark consists of an elastic quasi-static axisymmetric problem. The second benchmark is a 2D dynamic, elasto-plastic, vertical cut problem.

### 3.1 Axisymmetric benchmark

The first benchmark is similar to that presented by Naylor [19] and Mar and Hicks [18] to explore stress recovery. It consists of a hollow cylinder which deforms due to an incremental pressure (Δp_{s}) applied on the internal boundary (s). The main benefit of this benchmark is that, unlike a 1D plane strain problem, the stresses inside the elements are not constant; moreover, they deviate from the real solution and, depending on the material point position, the deviation may be large or small.

_{i}to the inner wall (s) changes, and is equal to the distance between the cylinder axis and the nearest active node (this implies that r

_{i}remains constant until the boundary material point jumps to the next element). To enable the numerical (large strain) solution to be interpreted in terms of the analytical (small strain) solution, the methodology includes the following three features: (1) the applied pressure Δp

_{s}on the boundary (s) is applied to the outer nodes of the elements containing the outer most material points; (2) due to the new location of the inner wall, Δp

_{s}is re-evaluated as Δp

_{s}(r

_{i}) = A/r

_{i}

^{2}+ 2C, where A and C are constants associated with the initial geometry and boundary conditions of the benchmark, as shown in Fig. 3 (a description of the analytical solution and the constants A and C are presented in “Appendix A2”); (3) instead of accumulated stresses, incremental stresses at the material points are used throughout the analysis. These three features ensure that the incremental stress at the material points, for an arbitrary position of the cylinder wall, can be compared to the analytical stress related to the original geometry of the cylinder.

The inner (initial) and outer cylinder boundaries are located at r_{i} = 0.5 m and r_{e} = 1.5 m, respectively. The cylinder domain is discretised by elements of dimension Δr = Δy = 0.20 m, and each element initially contains four material points equally spaced. The elastic properties are Young’s modulus, E = 1000 kPa, and Poisson’s ratio, ν = 0.30. The initial applied pressure increment is Δp_{s} = 100 kPa, and A and C are 19.56 kN and 10.87 kPa, respectively.

_{1}are plotted and compared to the analytical solution over 25 Δp

_{s}increments. It is evident that the stress invariants can deviate strongly from the analytical solution.

### 3.2 Vertical cut benchmark

_{p}= 12 kPa, the residual cohesion is c

_{r}= 3 kPa, and the softening modulus is H

_{s}= − 30 kPa. At the left boundary, the nodes are partly fixed to avoid displacement in the horizontal direction, whereas the nodes are fully fixed at the bottom boundary. The initial stresses in the domain are generated by fixing the locations of the material points and applying gravity loads until the internal and external forces are in equilibrium. After equilibrium is reached, the material points are released and deformation takes place.

## 4 Oscillations in MPM

The MPM technique can be seen as an FE stepwise procedure, in which the integration points (now called material points) move together with the mesh, but keep their new positions while the mesh returns to its original position. This allows the simulation of large deformations since extreme distortion of the mesh is avoided, although the process is found to cause stress oscillations. There are a number of contributing factors causing these oscillations, which are investigated below.

### 4.1 Stress recovery

_{L}) is different from that across the quadratic element (σ

_{Q}), and that both are different from the analytical stress (σ

_{A}). However, the linear and quadratic stresses (σ

_{L}and σ

_{Q}, respectively) match the analytical solution exactly at the Gauss point locations. This means that, depending on the position of the material point, the recovered stresses can be either higher or lower than the analytical stresses, as illustrated in Fig. 4.

### 4.2 Nodal integration using SF gradients

The nodal integrations of **F**^{int} and **K** are performed using SF gradients and the material point positions. However, considering that the SF gradients used in MPM are bi-linear (linear elements) and discontinuous, and that the material point positions change each time step, the resulting nodal values are inaccurate, especially if material points cross element boundaries. Next, a description of the SF gradients in MPM and the consequences of using them are presented.

#### 4.2.1 2D bi-linear shape functions

_{1}and E

_{2}(Fig. 10a). The SFs and SF gradients in both directions of node 5 are shown in Fig. 10b–d, respectively. Figure 10b shows that the SFs are continuous between elements, while Fig. 10d shows that the vertical SF gradient is continuous between elements in the horizontal direction and constant in the vertical direction. On the other hand, Fig. 10c shows that the horizontal SF gradients at the inter-element boundary are discontinuous, and that they decrease in the vertical direction.

#### 4.2.2 Integration of the internal forces **F**^{int} and stiffness **K**

Using SF gradients in the integration of any variable (i.e. **F**^{int} and **K**) results in an inadequate nodal distribution, whereas, if regular SFs are used, the nodal distribution is smoother (**M** and **F**^{ext}). Moreover, two differences should be noticed between the integration of **F**^{int} and **K**. The first is that, to integrate **F**^{int}, the strain–displacement matrix (**B**) is used once (Eq. 7), whereas the element stiffness is computed using both **B** and its transpose **B**^{T} (Eq. 5). The second is that to integrate **F**^{int}, the stresses of the material points are used, whereas to integrate **K** the elastic properties of the material points are used. The significance of this is that the elastic properties are constant throughout the analysis, whereas the material point stresses change during the analysis, causing possible accumulation of errors.

_{x}and K

_{y}) using two different material point distributions, are computed for nodes 1–5 of the plane strain finite element mesh shown in Fig. 11. In both cases the material points are equally distributed inside the elements; in the first case (Fig. 11a) the material points are located inside each element, whereas in the second case (Fig. 11b) the material points have moved and some are located at the inter-element boundaries. After the movement, the material points are still located inside their original element, except for material points a-d which have crossed the boundary by an infinitesimal distance. Stress components of σ

_{x}= σ

_{y}= − 1.0 MPa and σ

_{xy}= 0, a Young’s modulus of E = 1.0 kPa and a Poisson’s ratio of ν = 0 for each material point have been considered, while the distance between the nodes is 1 m and the material point weights are equal to 1.

In Fig. 11c, d, the vertical internal force is equal to zero in both cases. The force is unchanged because the horizontal displacement of the material points does not affect the values of the vertical SF gradients, and equals zero because the internal vertical forces on both sides of the nodes are the same but with an opposite sign. However, the distribution of the horizontal internal force is highly inaccurate due to the material point crossing the element boundary and the discontinuity of the horizontal SF gradients (Fig. 11d). When integrating the nodal stiffness, the horizontal and vertical stiffnesses are initially similar (Fig. 11e). However, as the material points cross an element boundary (Fig. 11f), the inaccuracies are evident again, although they are smaller than those of the internal forces. This is because the product **BB**^{T} returns positive nodal values, so avoiding the change in sign of the SF gradients.

### 4.3 Nodal integration of the mass **M** and external forces **F**^{ext} using SFs

**M**and

**F**

^{ext}is performed using SFs rather than SF gradients, so that discontinuities between elements do not occur. In this example, only the external forces caused by gravity are considered. Since a lumped form of the mass matrix is used, and also because of the partition of unity of SFs, any initial distribution of material points inside the elements results in the same nodal mass (or external force), as long as the distribution is symmetrical. As an example, Fig. 12 shows two different material point distributions inside an element, but the nodal mass and nodal external forces are the same in both cases.

**M**for the same problem as in Fig. 11. It is clear that the movement of material points and the crossing of nodes does not cause any trouble for the nodal integration because of the continuity of the SFs. Also, since the integration of

**F**

^{ext}is performed in a similar manner to

**M**, the distribution would be similar to the one in Fig. 13.

### 4.4 Plastic stress redistribution

The stress oscillation caused by the plastic stress redistribution is an extension of the oscillations explained in the previous sections. As the stresses exceeding the yield surface are integrated as a new external force computed with SF gradients, additional oscillations comparable to the **F**^{int} oscillations are introduced. Moreover, oscillating stresses could cause some points to yield spuriously, leading to an unrealistic system behaviour.

## 5 Improvements to reduce stress oscillations

### 5.1 GIMP

_{ip}) and its gradient (∇S

_{ip}) in one dimension are computed as

_{p}is the material point volume, Ω is the problem domain, Ω

_{p}is the material point support domain, i is the node, and χ

_{p}is the characteristic function delimiting the area of influence of the material point and is given as

The GIMP SFs in 2D and 3D are computed as products of the 1D GIMP SF in each direction; that is, \( {\rm S}_{\rm i} ({\rm x}) = {\rm S}_{\rm ip}^{1} ({\rm x}_{1} ) \cdot {\rm S}_{\rm ip}^{2} ({\rm x}_{2} ) \) in 2D and \( {\rm S}_{\rm i} ({\rm x}) = {\rm S}_{\rm ip}^{1} ({\rm x}_{1} ) \cdot {\rm S}_{\rm ip}^{2} ({\rm x}_{2} ) \cdot {\rm S}_{\rm ip}^{3} ({\rm x}_{3} ) \) in 3D, where \( {\rm S}_{\rm ip}^{\rm k} \) is the 1D GIMP SF in the k-direction. An additional advantage of including a support domain is that the material boundary is explicitly defined, and can be used to apply boundary conditions.

### 5.2 Modified integration weights

### 5.3 Double mapping (DM)

**D**

_{i}is the elastic matrix at node i, and

**D**

_{p}is the elastic matrix of material point p.

**D**

_{g}is the elastic matrix at the Gauss point, N

_{i}(

**x**

_{g}) is the nodal SF evaluated at the Gauss points, and nn is the number of nodes of the element. By substituting Eq. 16 into Eq. 17,

**D**

_{g}is obtained as

^{FE}is the weight associated with Gauss point g (as in FEM).

### 5.4 DM-GIMP (DM-G)

As shown in Fig. 15a, for the initial configuration of material points, the nodal stiffness distributions remain the same for both techniques, because at this position the MPM and GIMP SFs and SF gradients are the same. With the movement of the material points (Fig. 15b), the nodal stiffness computed with GIMP decreases, as the GIMP SF gradients drop to zero at the inter-element boundaries (as shown in Fig. 14). In addition, the contribution of material points in neighbouring elements is not capable of compensating for this drop. This would be the case for other methods, including DDMP and CDPI, that have this same characteristic.

_{ip*}) are similarly created as regular GIMP SFs, but the influence of the material point support domain affects only the nodal FE SF in a single element rather than contributing to all contiguous elements. In Fig. 16, an illustration of the development of regular and local GIMP shape functions of a node is shown.

_{ip*}is the local GIMP SF of node i evaluated at the material point position, and smp is the number of material points with a support domain inside the element. The algorithm to compute the stiffness matrix using DM and DM-G is given in “Appendix C”, together with a study of the computational performance.

### 5.5 Composite material point method (CMPM)

^{2}shape functions are shown in 1D, in which each shape function N

^{2}envelopes the local element plus the neighbouring elements.

^{2}shape functions is computed as

_{j}is the local coordinate of the N

_{i}

^{2}shape function, and ξ

_{m}is the local coordinate of the remaining nodes. Solving Eq. 21 for each node, the CMPM shape functions for an element with two neighbours are

## 6 Testing of proposed techniques

^{2}and ν = 0.30. Figure 20 presents the stiffness computed using regular MPM and DM and the results are compared with the FEM stiffness, computed using four Gauss integration points (K

_{mag}= 3263.57 kN/m

^{2}). In addition, the stiffness using the modified integration weights (W*) and Gauss mapping (GM) separately (the two components of DM) are shown to highlight their comparative effects. Since the material points remain equally distributed after rotation, the stiffness of the domain should not change (i.e. be mesh independent). Finally, a further test is performed using two materials, by considering the properties of material points below line A–A′ to be E = 1500 kN/m

^{2}and ν = 0.25.

Theoretically, the stiffness of the domain should be independent of the rotation of the field of material points, and should be equal to the FEM stiffness before rotation (for the case with one material). As can be observed in Fig. 20, the stiffness obtained using regular MPM is not accurate and improvements are needed. After including the modified integration weight (W*), which accounts for a varying amount of material points per element, the stiffness distribution oscillates, although with a different spatial pattern than in regular MPM. Using GM the oscillation also persists, as the number of material points per cell is still incorrect, but it is less than in regular MPM because it helps to reduce errors due to material point position. It is noted that including W* or GM separately is unable to fix the stiffness oscillation, and that the spatial distribution is almost opposite in pattern, i.e. where high values occur in GM, low values occur in W*, and vice versa. Using DM, i.e. combining GM and W*, the stiffness oscillation is reduced significantly, as it accounts for both the material point position and the number of material points per element. Moreover, the transition is smooth over the elements when two materials are used.

Relative differences in stiffness magnitude between FEM and other methods for homogeneous material

Method | Maximum increase (%) | Maximum decrease (%) |
---|---|---|

Regular MPM | 5.51 | − 7.38 |

W | 5.93 | − 8.88 |

GM | 4.75 | − 5.63 |

DM | 2.23 | − 2.38 |

GIMP | 0.0 | − 33.95 |

DM-G | 0.21 | − 0.39 |

## 7 Benchmark problems including improvements

The benchmark problems introduced in Sect. 3 are now re-analysed using the improvements described in Sect. 5.

### 7.1 Axisymmetric benchmark

_{1}(over 25 load steps) are shown, comparing the stresses obtained using normal MPM (as shown in Fig. 4), DM and DM-CMPM (DM-C). As can be seen, there is a significant increase in the accuracy of the stresses recovered using the DM technique, due to the improved stress recovery and stiffness integration. Moreover, if CMPM is included in the analysis, the stress oscillation reduces still further to give stresses close to the analytical solution.

_{i}= r

_{mp1}− lp as in Fig. 24. Then, the applied pressure Δp

_{s}is distributed linearly to the nodes of the boundary element based on proximity.

### 7.2 Vertical cut benchmark

**F**

^{int}magnitude is shown, once again comparing regular MPM and DM-GC. Analogous to Eq. 25, the magnitude of the nodal internal force is computed as

**F**

^{int}is obtained. Using GIMP, the oscillation caused by the material points crossing cell boundaries is reduced. Furthermore, by including CMPM, the recovered stresses are improved, reducing the oscillation caused by the stress recovery position.

As can be seen from previous figures, the oscillation of material point stresses, nodal stiffness and internal nodal forces are reduced significantly using DM-GC. Plots for the nodal mass and external nodal forces are not included in the results, since the oscillation for both MPM and DM-GC is small.

Figure 31 shows the p-q stress paths at the 3 points, as computed using both techniques, as well as the initial position of the yield surface for a Von Mises material (F_{VM}). It is seen that, for material point A, both techniques give reasonable results; this is because the bottom of the domain is fully fixed, so that the material point does not move much throughout the analysis. For material points B and C, if regular MPM is used (Fig. 31b, c), the oscillations are extreme. It is evident that were a constitutive model different from Von Mises to be used, in which plasticity does not depend only on the deviatoric stress, regular MPM would not perform well. On the other hand, using DM-GC, the stress path appears to be well-behaved (Fig. 31e, f), with only some small oscillations.

Summary of advantages and disadvantages of the methods studied

Method | Oscillations of nodal | Oscillations of | Stress recovery | ||
---|---|---|---|---|---|

MP position | MP CEB | MP position | MP CEB | – | |

MPM | Inaccurate distribution of internal forces | Large oscillations due to discontinuity of SF gradients | Increase if MP is close to the node, and decrease in some directions if MP is far from the node | Accumulation or reduction of MPs causes an increase or reduction of stiffness | Poor |

GIMP | Decrease of oscillation by keeping continuity of SF gradients | Imbalance disappears because of continuity of SF gradients | Large stiffness oscillation | Stiffness vanishes if MP is at element boundary | Continuous between elements |

DM-MPM | Inaccurate distribution of internal forces | Large imbalance due to discontinuity of SF gradients | Stiffness oscillation reduces | Stiffness oscillation reduces significantly by using the modified weighting value W* | Poor |

DM-G | Decrease of oscillation by keeping continuity of SF gradients | Imbalance disappears because of continuity of SF gradients | Stiffness oscillation reduces significantly | Stiffness oscillation reduces significantly | Continuous between elements |

DM-GC | Decrease of oscillation by keeping continuity of SF gradients | Imbalance disappears because of continuity of SF gradients | Stiffness oscillation reduces significantly | Stiffness oscillation reduces significantly | Highly improved by increasing the solution domain and SF order |

## 8 Conclusion

MPM is a technique that is able to handle problems involving large deformations, since material properties and the body geometry are no longer attached to a mesh. Unfortunately, the use of regular bi-linear finite element shape functions causes significant oscillations when integrating internal forces and stiffness, decreasing the accuracy of the simulations. Moreover, the grid crossing of a material point between elements and poor stress recovery create additional oscillations. A series of improvements, both novel and building upon the work of others, have been studied and combined to obtain an almost oscillation free version of MPM. It has been shown that GIMP reduces the errors caused by grid crossing, but integration using SF gradients, shown via an example using the stiffness matrix, is inaccurate due to the use of SF gradients that drop to zero at the inter-element boundaries. Using GIMP together with a double mapping integration procedure significantly reduces the stiffness matrix oscillation. Also, it has been proven that CMPM increases the accuracy of the stresses computed at the material points compared to typical MPM and GIMP. These techniques combined (termed DM-GC) increases considerably the accuracy of the MPM simulations. Moreover, since it has been observed that DM performs well when using typical finite element shape functions, and better still when using GIMP shape functions, the combination of DM with other C1-continuous methods, such as CPDI, B-spline MPM or DDMP, is a possibility which can be studied in the future. The DM and DM-G methods have the benefit of being able to be implemented implicitly or explicitly with typical elasto-plastic constitutive models.

## Notes

### Acknowledgements

The authors of this work express their gratitude to the Mexican National Council for Science and Technology (CONACYT) for financing this work through the Grant Number 409778.

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