Comparison of an h- and hp-Adaptive Finite Element Solver for Chemo-Mechanically Coupled Battery Electrode Particles

Numerical investigations of mechanical stresses for phase transforming battery electrode materials on the particle scale are computationally highly demanding. The limitations are mainly induced by the strongly varying spatial and temporal scales of the underlying phase field model, which require an ultra fine mesh and time resolution, however, solely at specific stages in space and time. To overcome these numerical difficulties we present a general-purpose space and time adaptive solution algorithm based on an hp-adaptive finite element method and a variable-step, variable-order time integrator. At the example of a chemo-mechanical electrode particle model we demonstrate the computational savings gained by the hp-adaptivity. In particular, we compare the results to an h-adaptive finite element method and show the reduction of computational complexity

Simulation of the Deformation for Cycling Chemo-Mechanically Coupled Battery Active Particles with Mechanical Constraints

Next-generation lithium-ion batteries with silicon anodes have positive characteristics due to higher energy densities compared to state-of-the-art graphite anodes. However, the large volume expansion of silicon anodes can cause high mechanical stresses, especially if the battery active particle cannot expand freely. In this article, a thermodynamically consistent continuum model for coupling chemical and mechanical effects of electrode particles is extended by a change in the boundary condition for the displacement via a variational inequality. This switch represents a limited enlargement of the particle swelling or shrinking due to lithium intercalation or deintercalation in the host material, respectively. For inequality constraints as boundary condition a smaller time step size is need as well as a locally finer mesh. The combination of a primal-dual active set algorithm, interpreted as semismooth Newton method, and a spatial and temporal adaptive algorithm allows the efficient numerical investigation based on a finite element method. Using the example of silicon, the chemical and mechanical behavior of one- and two-dimensional representative geometries for a charge-discharge cycle is investigated. Furthermore, the efficiency of the adaptive algorithm is demonstrated. It turns out that the size of the gap has an significant influence on the maximal stress value and the slope of the increase. Especially in two dimension, the obstacle can cause an additional region with a lithium-poor phase

System Level Synthesis Beyond Finite Impulse Response Using Approximation by Simple Poles

Optimal linear feedback control design is valuable but challenging. The system level synthesis approach uses a reparameterization to expand the class of problems that can be solved using convex reformulations, among other benefits. However, to solve system level synthesis problems prior work relies on finite impulse response approximations that lead to deadbeat control, and that can experience infeasibility and increased suboptimality, especially in systems with large separation of time scales. This work develops a new technique by combining system level synthesis with a new approximation based on simple poles. The result is a new design method which does not result in deadbeat control, is convex and tractable, always feasible, can incorporate prior knowledge, and works well for systems with large separation of time scales. A general suboptimality result is provided which bounds the approximation error based on the geometry of the pole selection. The bound is then specialized to a particularly interesting pole selection to obtain a non-asymptotic convergence rate. An example demonstrates superior performance of the method.Comment: 25 page

Simulation of the Deformation for Cycling Chemo-Mechanically Coupled Battery Active Particles with Mechanical Constraints

Next-generation lithium-ion batteries with silicon anodes have positive characteristics due to higher energy densities compared to state-of-the-art graphite anodes. However, the large volume expansion of silicon anodes can cause high mechanical stresses, especially if the battery active particle cannot expand freely. In this article, a thermodynamically consistent continuum model for coupling chemical and mechanical effects of electrode particles is extended by a change in the boundary condition for the displacement via a variational inequality. This switch represents a limited enlargement of the particle swelling or shrinking due to lithium intercalation or deintercalation in the host material, respectively. For inequality constraints as boundary condition a smaller time step size is need as well as a locally finer mesh. The combination of a primal-dual active set algorithm, interpreted as semismooth Newton method, and a spatial and temporal adaptive algorithm allows the efficient numerical investigation based on a finite element method. Using the example of silicon, the chemical and mechanical behavior of one- and two-dimensional representative geometries for a charge-discharge cycle is investigated. Furthermore, the efficiency of the adaptive algorithm is demonstrated. It turns out that the size of the gap has an significant influence on the maximal stress value and the slope of the increase. Especially in two dimension, the obstacle can cause an additional region with a lithium-poor phase

Distributed and Constrained H2 \mathcal{H}_2 Control Design via System Level Synthesis and Dual Consensus ADMM

Design of optimal distributed linear feedback controllers to achieve a desired aggregate behavior, while simultaneously satisfying state and input constraints, is a challenging but important problem in many applications, including future power systems with weather-dependent renewable generation. System level synthesis is a recent technique which has been used to reparametrize the optimal control problem as a convex program. However, prior work is restricted to a centralized control design, which lacks robustness to communication failures and disturbances, has high computational cost and does not preserve data privacy of local controllers. The main contribution of this work is to develop a distributed solution to the previous optimal control problem, while incorporating agent-specific and globally coupled constraints in a non-conservative manner. To achieve this, it is first shown that the dual of this problem is a distributed consensus problem. Then, an algorithm is developed based on the alternating direction method of multipliers to solve the dual while recovering a primal solution, and a convergence certificate is provided. Finally, the method's performance is demonstrated on a test case of control design for distributed energy resources that collectively provide stability services to the power grid

Convergence of simple adaptive Galerkin schemes based on h − h/2 error estimators

We discuss several adaptive mesh-refinement strategies based on (h − h/2)-error estimation. This class of adaptivemethods is particularly popular in practise since it is problem independent and requires virtually no implementational overhead. We prove that, under the saturation assumption, these adaptive algorithms are convergent. Our framework applies not only to finite element methods, but also yields a first convergence proof for adaptive boundary element schemes. For a finite element model problem, we extend the proposed adaptive scheme and prove convergence even if the saturation assumption fails to hold in general

A high-quality annually laminated sequence from Lake Belau, Northern Germany: Revised chronology and its implications for palynological and tephrochronological studies

The annually laminated record of Lake Belau offers an exceptional opportunity to investigate with high temporal resolution Holocene environmental change, aspects of climate history and human impact on the landscape. A new chronology based on varve counts, 14C-datings and heavy metal history has been established, covering the last 9400 years. Based on multiple varve counting on two core sequences, the easily countable laminated section spans about 7850 varve years (modelled age range c. 9430 to 1630 cal. BP). Not all of the record is of the same quality but approximately 69% of the varves sequence is classified to be of high quality and only c. 5% of low quality. The new chronology suggests dates generally c. 260 years older than previously assumed for the laminated section of the record. The implications for the vegetation and land-use history of the region as well as revised datings for pollen stratigraphical events are discussed. Tephra analysis allowed the identification of several cryptotephra layers. New dates for volcanic eruptions are presented for the Lairg B event (c. 6848 cal. BP, 2s range 6930–6713 cal. BP), the Hekla 4 event (c. 4396 cal. BP, 2s range 4417–4266 cal. BP), and Hekla 3 eruption (c. 3095 cal. BP, 2s range 3120–3068 cal. BP)