An efficient mixed finite element formulation for 3D strain gradient elasticity

Mixed finite element formulations enable the use of simple continuous interpolation functions and standard finite element meshes for the numerical solution of (nonlocal) gradient elasticity problems. Yet, so far they are faced with the challenge of being costly due to the introduction of additional mixed solution variables. Furthermore, stability is not always observed in numerical benchmark tests, often necessitating stabilization terms. Since existing formulations lack second gradients in the weak form, consistency of the corresponding Euler–Lagrange equations with the surface flux equations of the original problem is not guaranteed, which does not always enable the application of arbitrary Neumann and Dirichlet boundary conditions of first and second order limiting practical applications. Therefore, in the present contribution a formulation is proposed, which is shown to be consistent for any kind of boundary conditions by adding suitable additional surface integrals. Furthermore, by incorporating the strains instead of the displacement gradient as a mixed variable and using simple interpolation functions that enable the condensation of the Lagrange multiplier, we arrive at an element that has less degrees of freedom than e.g. a standard P3 element for local elasticity. In numerical tests, it is shown that e.g. Hermite elements do not converge already in a simple benchmark problem, whereas the proposed formulation does not lack convergence, independent from the value of the length scale. In addition to that, the proposed element is shown to lead to a significantly reduced computing time compared to competitive formulations while maintaining advantageous convergence rates.