<p>The second-order cone linear complementarity problem (SOCLCP) can be used as a model of contact problems with friction. However, its standard formulation as a SOCLCP cannot be employed to large 3d problems because it requires a matrix inversion, which has a computational complexity of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$O(N_{uo}^3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msubsup> <mi>N</mi> <mrow> <mi mathvariant="italic">uo</mi> </mrow> <mn>3</mn> </msubsup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Therefore and here, inspired by the flexibility and structural simplicity of the well known MMS algorithms, we propose the Targeting-MMS algorithm for the SOCLCP model of 3d contact problems with friction. The computational complexity is reduced to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$O(N_{uo}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msubsup> <mi>N</mi> <mrow> <mi mathvariant="italic">uo</mi> </mrow> <mn>2</mn> </msubsup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> (or even to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$O(N_{uo})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msub> <mi>N</mi> <mrow> <mi mathvariant="italic">uo</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>). More importantly, we theoretically provide the optimal parameter for the new algorithm, which is a key point for a iteration method. Numerical results show that the Targeting-MMS algorithm with the proposed optimal parameter can save <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$85\%-90\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>85</mn> <mo>%</mo> <mo>-</mo> <mn>90</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> of the time overhead compared to the MMS algorithms and the semi-smooth Newton method, which are by far the state-of-the-art and commonly used methods for solving the SOCLCP.</p>

Targeting modulus-based matrix splitting iteration approach for the second-order cone linear complementarity model of three-dimensional frictional contact problems

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发表日期：

2026年1月9日

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Zhizhi Li,
Yimin Jin,
Huai Zhang

摘要

The second-order cone linear complementarity problem (SOCLCP) can be used as a model of contact problems with friction. However, its standard formulation as a SOCLCP cannot be employed to large 3d problems because it requires a matrix inversion, which has a computational complexity of $O(N_{uo}^3)$ $O (N_{uo}^{3})$ . Therefore and here, inspired by the flexibility and structural simplicity of the well known MMS algorithms, we propose the Targeting-MMS algorithm for the SOCLCP model of 3d contact problems with friction. The computational complexity is reduced to $O(N_{uo}^2)$ $O (N_{uo}^{2})$ (or even to $O(N_{uo})$ $O (N_{uo})$ ). More importantly, we theoretically provide the optimal parameter for the new algorithm, which is a key point for a iteration method. Numerical results show that the Targeting-MMS algorithm with the proposed optimal parameter can save $85\%-90\%$ $85 % - 90 %$ of the time overhead compared to the MMS algorithms and the semi-smooth Newton method, which are by far the state-of-the-art and commonly used methods for solving the SOCLCP.