A component-free Lagrangian finite element formulation for large strain elastodynamics

Abstract

Standard finite element formulation and implementation in solid dynamics at large strains usually relies upon and indicial-tensor Voigt notation to factorized the weighting functions and take advantage of the symmetric structure of the algebraic objects involved. In the present work, a novel component-free approach, where no reference to a basis, axes or components is made, implied or required, is adopted for the finite element formulation. Under this approach, the factorisation of the weighting function and also of the increment of the displacement field, can be performed by means of component-free operations avoiding both the use of any index notation and the subsequent reorganisation in matrix Voigt form. This new approach leads to a straightforward implementation of the formulation where only vectors and second order tensors in $${\mathbb {R}}^3$$ R 3 are required. The proposed formulation is as accurate as the standard Voigt based finite element method however is more efficient, concise, transparent and easy to implement.

Toward a local maximum‐entropy material point method at finite strain within a B‐free approach

“…Although the nodal internal virtual work and the tangent density matrix of the proposed B‐free MPM approach are, in principle, completely different from those derived in the standard Voigt‐based MPM formulation, no significant differences can be appreciated between both formulations from Figure 9(B). It is worth mentioning that this assertion has been validated by the authors in Stickle et al 27 under the FEM framework. It can be seen that both solutions tend to diverge on time, but respecting the amplitude of the oscillations.…”

“…Furthermore, in the aforementioned publication, no benchmark was provided to verify the performance of the B-free approach. In order to extend Planas's work, the authors of the present research have described the complete procedure using a Lagrangian description for the nonlinear elastodynamic problem, providing expressions for the Ciarlet 26 Neo-Hookean material and proposing several benchmarks, see Stickle et al 27 Some of them are also reproduced in the present research. The previously cited publications exploit the benefits of the B-free formulation under the FEM context.…”

“…The IBVP enunciated in the previous section, Subsection 2.2, describes the motion of a continuum within a set of partial differential equations (PDEs). These equations could be solved analytically for some simple problems, 27 but have to be solved numerically for most of real problems. The approach followed by numerical techniques, such as the MPM, is first establishing a weak form, which is equivalent to the original PDEs with their initial and boundary conditions, and then solving it numerically.…”

“…Working in the reference configuration has some interesting properties: it is the ideal framework to linearize the nonlinear problem due to the frame invariance of ; the conservation of mass is ensured by default; and any rate of a tensorial magnitude is objective by definition. Further discussion between B‐free structure and B‐based in the context of the FEM can be found in a previous research of the authors, see Stickle et al 27 As will be seen in the subsequent developments, to overcome the clear limitations of the Voigt algebra the authors' bet for the use of the standard tensor calculus in pursuit of simplicity. At this stage of the development, the spatial discretization and related linearization is introduced in Equations (18) and (19) to obtain a nonlinear system of algebraic equations.…”

The Local Maximum-Entropy Material Point Method

“…• Component-free linearization in the reference configuration [144] • Incremental Newmark-β Chapter 2…”