Consistent coupling of positions and rotations for embedding 1D Cosserat beams into 3D solid volumes

The present article proposes a mortar-type finite element formulation for consistently embedding curved, slender beams into 3D solid volumes. Following the fundamental kinematic assumption of undeformable cross-section s, the beams are identified as 1D Cosserat continua with pointwise six (translational and rotational) degrees of freedom describing the cross-section (centroid) position and orientation. A consistent 1D-3D coupling scheme for this problem type is proposed, requiring to enforce both positional and rotational constraints. Since Boltzmann continua exhibit no inherent rotational degrees of freedom, suitable definitions of orthonormal triads are investigated that are representative for the orientation of material directions within the 3D solid. While the rotation tensor defined by the polar decomposition of the deformation gradient appears as a natural choice and will even be demonstrated to represent these material directions in a $$L_2$$ L 2 -optimal manner, several alternative triad definitions are investigated. Such alternatives potentially allow for a more efficient numerical evaluation. Moreover, objective (i.e. frame-invariant) rotational coupling constraints between beam and solid orientations are formulated and enforced in a variationally consistent manner based on either a penalty potential or a Lagrange multiplier potential. Eventually, finite element discretization of the solid domain, the embedded beams, which are modeled on basis of the geometrically exact beam theory, and the Lagrange multiplier field associated with the coupling constraints results in an embedded mortar-type formulation for rotational and translational constraint enforcement denoted as full beam-to-solid volume coupling (BTS-FULL) scheme. Based on elementary numerical test cases, it is demonstrated that a consistent spatial convergence behavior can be achieved and potential locking effects can be avoided, if the proposed BTS-FULL scheme is combined with a suitable solid triad definition. Eventually, real-life engineering applications are considered to illustrate the importance of consistently coupling both translational and rotational degrees of freedom as well as the upscaling potential of the proposed formulation. This allows the investigation of complex mechanical systems such as fiber-reinforced composite materials, containing a large number of curved, slender fibers with arbitrary orientation embedded in a matrix material.

Impact of Flow Based Market Coupling on the European Electricity Markets

Flow Based Market Coupling is the target model for determining exchange capacities in the internal European Electricity Market. It has been in operation in Central Western Europe since 2015 and is scheduled to be extended to the wider Core region in the near future. Exchange capacities have a significant impact on market prices, exchanges and the energy mix, thus also determining the CO$${}_{2}$$ 2 footprint of electricity generation in the system. Stakeholders therefore need to develop an understanding for the impact of Flow Based Market Coupling and the parameter choice, like the minimum exchange capacities introduced in 2020, on the market outcome. This article presents a framework to model Flow Based Market Coupling and analyse the impact of different levels of regulatory induced minimum trading capacities as well as the effect of the extension towards the Core region. Electricity prices, exchange positions and the number and nature of binding constraints in the market results under different market coupling scenarios are investigated. The results show that increased level of minimum trading capacities in CWE market coupling decrease the German net export position by up to 7 TW h or 23%, while French exports increase by up to 10 TW h or 9%. The different transfer capacity in the scenarios induce a price difference of up to 13%. Increased exchange capacities allow for more base load generation with the corresponding effects for the CO$${}_{2}$$ 2 emissions of the system. The nature of coupling constraints is highly dynamic and dependent on the system state, which makes the suitability of static NTC values in energy system scenarios questionable.

Alternating Inertial and Overrelaxed Algorithms for Distributed Generalized Nash Equilibrium Seeking in Multi-Player Games

This paper investigates the distributed computation issue of generalized Nash equilibrium (GNE) in a multi-player game with shared coupling constraints. Two kinds of relatively fast distributed algorithms are constructed with alternating inertia and overrelaxation in the partial-decision information setting. We prove their convergence to GNE with fixed step-sizes by resorting to the operator splitting technique under the assumptions of Lipschitz continuity of the extended pseudo-gradient mappings. Finally, one numerical simulation is given to illustrate the efficiency and performance of the algorithm.

Partially distributed outer approximation

This paper presents a novel partially distributed outer approximation algorithm, named PaDOA, for solving a class of structured mixed integer convex programming problems to global optimality. The proposed scheme uses an iterative outer approximation method for coupled mixed integer optimization problems with separable convex objective functions, affine coupling constraints, and compact domain. PaDOA proceeds by alternating between solving large-scale structured mixed-integer linear programming problems and partially decoupled mixed-integer nonlinear programming subproblems that comprise much fewer integer variables. We establish conditions under which PaDOA converges to global minimizers after a finite number of iterations and verify these properties with an application to thermostatically controlled loads and to mixed-integer regression.