Stability Analysis of Decoupled Solution Strategies for Coupled Multi-field Problems – A General Framework

A coupled partial differential equation (PDE) system, stemming from the mathematical modelling of a coupled phenomenon, is usually solved numerically following a monolithic or a decoupled solution method. In spite of the potential unconditional stability offered by monolithic solvers, their usage for solving complex problems sometimes proves cumbersome. This has motivated the development of various partitioned and staggered solution strategies, generally known as decoupled solution schemes. To this end, the problem is broken down into several isolated yet communicating sub-problems that are independently advanced in time, possibly by different integrators. Nevertheless, using a decoupled solver introduces additional errors to the system and, therefore, may jeopardise the stability of the solution [1]. Consequently, to scrutinise the stability of the solution scheme becomes a pertinent step in proposing decoupled solution strategies. Here, we endeavour to present a practical stability analysis algorithm, which can readily be used to reveal the stability condition of numerical solvers. To illustrate its capabilities, the algorithm is then utilised for the stability analysis of solution schemes applied to multi variate coupled PDE systems resulting from the mathematical modelling of surface- and volume-coupled multi-field problems. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Experimental study of the effect of disk obstacle rotating with Different angular ratios on heat transfer and pressure drop in a pipe with turbulent flow

Journal of Thermal Analysis and Calorimetry - This article presents the effects of a circular disk obstacle with different angle ratios on heat transfer and pressure drop under a turbulent flow...     

Effect of overlapping cellulose nanofibrils and nanoclay layers on mechanical and barrier properties of spray-coated papers

Cellulose - This work evaluates the effect of spray-coating papers using cellulose nanofibrils (CNFs) and nanoclay (NC) on the mechanical and barrier properties for application such as reinforced...     

On the Stability Analysis of Decoupled Solution Schemes

Many problems in engineering, physics or other disciplines require an integrated treatment of coupled fields. These problems are characterised by a dynamic interaction among two or more physically or computationally distinct components, where the undergoing mathematical model commonly consists of a system of coupled PDE. Considerable progress has been made in the development of appropriate computational schemes to solve such coupled PDE systems. These attempts have resulted in various monolithic and decoupled numerical solution approaches. Despite the unconditional stability offered by implicit monolithic solution strategies, their use is not always recommended. The reason mainly lies in the complexity of the resulting system of equations and the limited flexibility in choosing appropriate time integrators for individual components. This has motivated the elaboration of tailored decoupled solution schemes, which follow the idea of splitting the problem into several sub-problems. But selection of the way of splitting can have a direct influence on the stability of the resulting solution algorithm. This necessitates the stability analysis of such an algorithm. Here, we introduce a general framework for the stability analysis of decoupled solution schemes. The approach is then used to study the stability behaviour of established decoupling strategies applied to typical volume- and surface-coupled problems, namely, coupled problems of thermoelasticity, porous media dynamics and structure-structure interaction. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the General Solution of Coupled Problems: Comparison of Monolithic and Partitioning Approaches

Coupled systems arise whenever there is a dynamic interaction between two or more functionally distinct components. The mathematical model for such a phenomenon consists of a system of coupled partial differential equations (PDE) in space and time, that has to be solved, either analytically or numerically, in order to describe or predict the response of the system under specific conditions. In spite of the natural accuracy of the analytical methods, they are less favourable due to their disability to treat more complex problems. Instead, different numerical schemes have been developed during the last decades, which are specialised to solve various coupled problems. These methods can be divided into two main categories, namely, monolithic and partitioned approaches. The main goal of this work is to study the general ideas behind monolithic and partitioned solution schemes for the pure differentially coupled systems. In the next step, the coupled problem of linear thermoelasticity is considered as benchmark example and the isothermal and isentropic operator-splitting schemes as two typical decoupling methods for this problem are presented. A canonical initial boundary-value problem has also been solved monolithically as well as by using the a.m. operator-splitting schemes and using the acquired results, the efficiency and stability of the methods are compared. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Staggered solution of fluid-porous-media interaction using the method of local Lagrange multipliers

Surface interaction among non-overlapping bulk-fluid and porous-medium bodies occurs in different situations, e. g., the interaction of blood with a blood vessel wall, a body of water with an earth dam structure, or acoustic waves with acoustic panels used in soundproofing. These are multi-field phenomena, comprising various surface- and volume-coupling mechanisms that should be reflected in the corresponding mathematical models. These models, together with appropriate initial and boundary values, assemble a coupled problem, the solution of which reveals the behaviour of the system under external excitations. The solution is commonly done numerically, following a monolithic or a decoupled approach. Here, the focus is on the latter. To design an efficient decoupled scheme, different types of coupling within the problem are addressed. These are the volume coupling between the degrees of freedom (DOF) within each subdomain, and the surface coupling between the DOF on the common boundaries. In particular, the latter constrains the feasible space of the solution of the problem. In this regard, local Lagrange multipliers (LLM) are employed to reformulate the problem in an unconstrained form. Unlike other domain decomposition methods which are based on using global Lagrange multipliers, the LLM method yields a complete separation of the subdomains and, consequently, facilitates parallel solution of the sub-problems. Moreover, within the subdomains, the penalty method is used to decouple pressure from other DOF. This procedure, on the one hand, reduces the size of the problem that should be solved at the interface and, on the other hand, removes the burden of using mixed finite elements within the subsystems. In the next step, the stability behaviour of the resulting staggered approach is analysed, and the unconditional stability of the method is established. Finally, the method is employed to solve a benchmark example, and using the numerical results, the reliability of the outcomes of the stability analysis is investigated. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupled Multi-field and Multi-rate Problems – Numerical Solution and Stability Analysis

The numerical solution of coupled differential equation systems is usually done following a monolithic or a decoupled algorithm. In contrast to the holistic monolithic solvers, the decoupled solution strategies are based on breaking down the system into several subsystems. This results in different characteristics of these families of solvers, e. g., while the monolithic algorithms provide a relatively straight-forward solution framework, unlike their decoupled counterparts, they hinder software re-usability and customisation. This is a drawback for multi-field and multi-rate problems. The reason is that a multi-field problem comprises several subproblems corresponding to interacting subsystems. This suggests exploiting an individual solver for each subproblem. Moreover, for the efficient solution of a multi-rate problem, it makes sense to perform the temporal integration of each subproblem using a time-step size relative to its evolution rate. Nevertheless, decoupled solvers introduce additional errors to the solution and, thus, they must always be accompanied by a thorough stability analysis. Here, tailored solution schemes for the decoupled solution of multi-field and multi-rate problems are proposed. Moreover, the stability behaviour of the solutions obtained from these methods are studied. Numerical examples are solved and the reliability of the outcome of the stability analysis is investigated. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)