<p>The material point method (MPM) is a particle method, suitable for large-deformation analysis. Bodies are represented by particles, the so-called material points. These carry all information such as mass, velocity, position and deformation state, to name just a few. With this in hand, the Lagrangian representation of the body is combined with a computational background grid in an Eulerian sense in order to solve the differential equations of interest. Since the grid does not keep track of history data it is reset at the beginning of each time step using the same geometry throughout the simulation. As a result, mesh distortion is diminished, enabling large deformations throughout the simulation. While using <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40571_2025_1026_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$C^0$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation>-continuous shape functions, stress oscillations are likely to occur. One way of solving this issue is the application of higher-order shape functions. In this contribution, focus is on <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40571_2025_1026_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$C^0$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation>-continuous shape functions in combination with the grid-shift technique in order to reduce stress oscillations. Here, the initial placement of the grid nodes is altered, i.e., the grid is translated in space at the beginning of each time step. This method is realized by randomly shifting the origin of the grid, resulting in no computational overhead and a straightforward implementation. Comparisons of four numerical examples are analyzed using the standard fixed grid approach and the grid-shift technique with respect to oscillations in the stress field, alternative application of boundary conditions and energy conservation.</p>

Reduction of stress oscillations in the material point method based on the random grid-shift technique

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2025年7月19日

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M. Koßler,
S. Maassen,
R. Niekamp,
J. Schröder

摘要

The material point method (MPM) is a particle method, suitable for large-deformation analysis. Bodies are represented by particles, the so-called material points. These carry all information such as mass, velocity, position and deformation state, to name just a few. With this in hand, the Lagrangian representation of the body is combined with a computational background grid in an Eulerian sense in order to solve the differential equations of interest. Since the grid does not keep track of history data it is reset at the beginning of each time step using the same geometry throughout the simulation. As a result, mesh distortion is diminished, enabling large deformations throughout the simulation. While using $C^0$ $C^{0}$ -continuous shape functions, stress oscillations are likely to occur. One way of solving this issue is the application of higher-order shape functions. In this contribution, focus is on $C^0$ $C^{0}$ -continuous shape functions in combination with the grid-shift technique in order to reduce stress oscillations. Here, the initial placement of the grid nodes is altered, i.e., the grid is translated in space at the beginning of each time step. This method is realized by randomly shifting the origin of the grid, resulting in no computational overhead and a straightforward implementation. Comparisons of four numerical examples are analyzed using the standard fixed grid approach and the grid-shift technique with respect to oscillations in the stress field, alternative application of boundary conditions and energy conservation.