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A new shape optimization approach for fracture propagation

Tim Suchan, Kathrin Welker, Winnifried Wollner

TL;DR

This work presents a shape-optimization approach to quasi-static brittle fracture that eliminates the phase-field length-scale parameter by formulating fracture evolution as a shape optimization problem on a curve-based fracture within a two-dimensional domain. Using the Steklov–Poincaré metric, the authors compute shape gradients from a weak formulation of linear elasticity and incorporate a mesh-extension strategy via an Eikonal-based normal field to stabilize updates, including a mild volume regularization to control domain growth. The method is tested on tension and shear benchmarks, demonstrating fracture initiation and propagation with results that are qualitatively aligned with existing phase-field studies, though some load–displacement curves show deviations likely due to algorithmic choices and initial tip geometry. The study highlights the potential of geometry-driven fracture propagation modeling and outlines directions for future refinement, such as sensitivity to initial shape and the choice of deformation bilinear form for better quantitative agreement and robustness.

Abstract

Within this work, we present a novel approach to fracture simulations based on shape optimization techniques. Contrary to widely-used phase-field approaches in literature the proposed method does not require a specified 'length-scale' parameter defining the diffused interface region of the phase-field. We provide the formulation and discuss the used solution approach. We conclude with some numerical comparisons with well-established single-edge notch tension and shear tests.

A new shape optimization approach for fracture propagation

TL;DR

This work presents a shape-optimization approach to quasi-static brittle fracture that eliminates the phase-field length-scale parameter by formulating fracture evolution as a shape optimization problem on a curve-based fracture within a two-dimensional domain. Using the Steklov–Poincaré metric, the authors compute shape gradients from a weak formulation of linear elasticity and incorporate a mesh-extension strategy via an Eikonal-based normal field to stabilize updates, including a mild volume regularization to control domain growth. The method is tested on tension and shear benchmarks, demonstrating fracture initiation and propagation with results that are qualitatively aligned with existing phase-field studies, though some load–displacement curves show deviations likely due to algorithmic choices and initial tip geometry. The study highlights the potential of geometry-driven fracture propagation modeling and outlines directions for future refinement, such as sensitivity to initial shape and the choice of deformation bilinear form for better quantitative agreement and robustness.

Abstract

Within this work, we present a novel approach to fracture simulations based on shape optimization techniques. Contrary to widely-used phase-field approaches in literature the proposed method does not require a specified 'length-scale' parameter defining the diffused interface region of the phase-field. We provide the formulation and discuss the used solution approach. We conclude with some numerical comparisons with well-established single-edge notch tension and shear tests.
Paper Structure (7 sections, 8 equations, 4 figures)

This paper contains 7 sections, 8 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem description
  3. Shape optimization approach
  4. Numerical results
  5. Tension test.
  6. Shear test.
  7. Conclusion

Figures (4)

  • Figure 1: Sketch of the hold-all domain $D$.
  • Figure 2: Sketches of the computational domain $\varOmega$ with boundaries and boundary conditions for the tension test (left) and for the shear test (right).
  • Figure 3: Fracture propagation for the tension test.
  • Figure 5: Fracture propagation for the shear test.