An adaptive hybrid time‐stepping scheme for highly non‐linear strongly coupled problems

This paper deals with the design and implementation of an adaptive hybrid scheme for the solution of highly non-linear, strongly coupled problems. The term ‘hybrid’ refers to a composite time stepping scheme where a controller decides whether a monolithic scheme or a fractional step (splitting) scheme is appropriate for a given time step. The criteria are based on accuracy and efficiency. The key contribution of this paper is the development of a framework for incorporating error criteria for stepsize selection and a mechanism for choosing from splitting or monolithic possibilities. The resulting framework is applied to silylation, a highly non-linear, strongly coupled problem of solvent diffusion and reaction in deforming polymers. Numerical examples show the efficacy of our new hybrid scheme on both two- and three-dimensional silylation simulations in the context of microlithography. Copyright © 2005 John Wiley & Sons, Ltd.

[1]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[2]  J. Verwer,et al.  Analysis of operator splitting for advection-diffusion-reaction problems from air pollution modelling , 1999 .

[3]  J. C. Simo,et al.  Variational and projection methods for the volume constraint in finite deformation elasto-plasticity , 1985 .

[4]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[5]  Keith Miller,et al.  Design and Application of a Gradient-Weighted Moving Finite Element Code I: in One Dimension , 1998, SIAM J. Sci. Comput..

[6]  In Lee,et al.  Finite element analysis of transient thermoelastic behaviors in disk brakes , 2004 .

[7]  Bernhard A. Schrefler,et al.  Standard staggered and staggered Newton schemes in thermo-hydro-mechanical problems , 1997 .

[8]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

[9]  Tarek I. Zohdi,et al.  Modeling and simulation of a class of coupled thermo-chemo-mechanical processes in multiphase solids , 2004 .

[10]  Francisco Armero,et al.  A new unconditionally stable fractional step method for non‐linear coupled thermomechanical problems , 1992 .

[11]  O. C. Zienkiewicz,et al.  Unconditionally stable staggered solution procedure for soil-pore fluid interaction problems , 1988 .

[12]  Sanjay Govindjee,et al.  Coupled stress-diffusion: Case II , 1993 .

[13]  Prashanth K. Vijalapura,et al.  Fractional step methods for index-1 differential-algebraic equations , 2005 .

[14]  Paola Causin,et al.  A simulation system based on mixed-hybrid finite elements for thermal oxidation in semiconductor technology , 2004 .

[15]  Prashanth K. Vijalapura,et al.  Numerical simulation of coupled‐stress case II diffusion in one dimension , 2003 .

[16]  Noreen L. Thomas,et al.  A theory of case II diffusion , 1982 .

[17]  Detlef Kuhl,et al.  Coupled chemo-mechanical deterioration of cementitious materials Part II: Numerical methods and simulations , 2004 .

[18]  Noreen L. Thomas,et al.  Case II swelling of PMMA sheet in methanol , 1978 .

[19]  A. Arshak,et al.  Top surface imaging lithography processes for I-line resists using liquid-phase silylation , 2002, 2002 23rd International Conference on Microelectronics. Proceedings (Cat. No.02TH8595).

[20]  Khalil Arshak,et al.  PRIME process with Shipley SPR505A resist—simulations and experiments , 2002 .

[21]  Stefan Hartmann,et al.  Remarks on the interpretation of current non‐linear finite element analyses as differential–algebraic equations , 2001, International Journal for Numerical Methods in Engineering.

[22]  David Dureisseix,et al.  A LATIN computational strategy for multiphysics problems: application to poroelasticity , 2003 .

[23]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

[24]  Francisco Armero,et al.  Formulation and finite element implementation of a multiplicative model of coupled poro-plasticity at finite strains under fully saturated conditions , 1999 .

[25]  T. Zohdi.  An adaptive‐recursive staggering strategy for simulating multifield coupled processes in microheterogeneous solids , 2002 .

[26]  Dimitrios Stamatialis,et al.  Investigation of Case II Diffusion Behavior. 1. Theoretical Studies Based on the Relaxation Dependent Solubility Model , 2002 .

[27]  S. Sloan,et al.  BIOT CONSOLIDATION ANALYSIS WITH AUTOMATIC TIME STEPPING AND ERROR CONTROL PART 1: THEORY AND IMPLEMENTATION , 1999 .

[28]  A. Argon,et al.  A mechanistic model of case-II diffusion of a diluent into a glassy polymer , 1999 .

[29]  Mark A. Hartney.  Modeling of positive‐tone silylation processes for 193‐nm lithography , 1993 .

[30]  Giulio C. Sarti,et al.  Coupled deformation and mass-transport processes in solid polymers , 1990 .

[31]  Bernard Schrefler,et al.  On convergence conditions of partitioned solution procedures for consolidation problems , 1993 .