A new conservative/dissipative time integration scheme for nonlinear mechanical systems

Abstract

We present a conservative/dissipative time integration scheme for nonlinear mechanical systems. Starting from a weak form, we derive algorithmic forces and velocities that guarantee the desired conservation/dissipation properties. Our approach relies on a collection of linearly constrained quadratic programs defining high order correction terms that modify, in the minimum possible way, the classical midpoint rule so as to guarantee the strict energy conservation/dissipation properties. The solution of these programs provides explicit formulas for the algorithmic forces and velocities which can be easily incorporated into existing implementations. Similarities and differences between our approach and well-established methods are discussed as well. The approach, suitable for reduced-order models, finite element models, or multibody systems, is tested and its capabilities are illustrated by means of several examples.

Structure‐preserving space‐time discretization of large‐strain thermo‐viscoelasticity in the framework of GENERIC

“…We perform a change of variables to transform the inelastic bracket (30) to the present setting which is based on functionals of the form (4). In particular, to link the current density functions a( , F, p, , C −1 p ) to those in ( 29), we express the generalized thermodynamic variable ∈ { , , u} in terms of the temperature by inverting relation (10) to get…”

“…Associated with the preservation of the geometric structure excellent long-term performance and numerical stability have been reported. In the realm of Langragian/Hamiltonian mechanics, many geometric integrators can be gathered around two classes: Variational integrators (see, e.g., Marsden and Ratiu, 3 Lew et al, 4 and Lall and West 5 for more details) and energy-momentum integrators (see, e.g., LaBudde and Greenspan, 6,7 Gonzalez, 8 or Betsch 9 for a comprehensive overview of previous developments), and energy-decaying variants thereof; see Gebhardt et al 10 and references therein. Numerous attempts have been made to extend structure-preserving schemes to the domain of non-conservative mechanical systems, such as port-Hamiltonian systems, 11 viscoelasticity, 12,13 elastoplasticity, 14 and thermo-viscoelasticity.…”

On a nonlinear rod exhibiting only axial and bending deformations: mathematical modeling and numerical implementation

“…In a recent publication [16], we showed that the hybrid combination of the midpoint and trapezoidal rules can render robust second-order, implicit integration methods that at least exactly preserve the linear and angular momenta. Therefore, this is the approach that we are going to pursue in the following.…”

The Rotating Rigid Body Model Based on a Non-twisting Frame

“…A comprehensive description of such ideas in the context of general nonholonomic dynamics is to be found in Celledoni et al (2019). The specialization of approaches based on more elaborated conservative/dissipative integration schemes like Gebhardt et al (2020) is possible as well in this context, but falls outside the scope of the current work and therefore, not addressed here. Lastly, be aware that the proposed model should not be understood as an alternative formulation to the wellknown standard rigid body model but as a different mechanical problem that possesses interesting properties, which can find applications in fields like multibody systems, n-body problems on manifolds, computer graphics and ballistics among others.…”