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Computing an upper bound on contact stress with surrogate duality

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Abstract

We present a method for computing an upper bound on the contact stress of elastic bodies. The continuum model of elastic bodies with contact is first modeled as a constrained optimization problem by using finite elements. An explicit formulation of the total contact force, a fraction function with the numerator as a linear function and the denominator as a quadratic convex function, is derived with only the normalized nodal contact forces as the constrained variables in a standard simplex. Then two bounds are obtained for the sum of the nodal contact forces. The first is an explicit formulation of matrices of the finite element model, derived by maximizing the fraction function under the constraint that the sum of the normalized nodal contact forces is one. The second bound is solved by first maximizing the fraction function subject to the standard simplex and then using Dinkelbach’s algorithm for fractional programming to find the maximum—since the fraction function is pseudo concave in a neighborhood of the solution. These two bounds are solved with the problem dimensions being only the number of contact nodes or node pairs, which are much smaller than the dimension for the original problem, namely, the number of degrees of freedom. Next, a scheme for constructing an upper bound on the contact stress is proposed that uses the bounds on the sum of the nodal contact forces obtained on a fine finite element mesh and the nodal contact forces obtained on a coarse finite element mesh, which are problems that can be solved at a lower computational cost. Finally, the proposed method is verified through some examples concerning both frictionless and frictional contact to demonstrate the method’s feasibility, efficiency, and robustness.

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Acknowledgments

We would like to acknowledge the generous support for this research provided by the National Natural Science Foundation of China under grant number 11172209, and the first author would like to acknowledge the China Scholarship Council for generously supporting his visit to UC Berkeley.

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Authors and Affiliations

  1. Department of Computer Science, Tianjin University of Technology and Education, Tianjin, 300222, China

    Zhaocheng Xuan

  2. Department of Mechanical Engineering, University of California, Berkeley, CA, 94720-1740, USA

    Panayiotis Papadopoulos

Authors
  1. Zhaocheng Xuan
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  2. Panayiotis Papadopoulos
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Correspondence to Zhaocheng Xuan.

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Xuan, Z., Papadopoulos, P. Computing an upper bound on contact stress with surrogate duality. Comput Mech 58, 171–183 (2016). https://doi.org/10.1007/s00466-016-1287-x

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  • DOI: https://doi.org/10.1007/s00466-016-1287-x

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