General Single Step Single Solve integration algorithm

Direct linear or nonlinear explicit or implicit time integration of structural dynamics problems

% General Single Step Single Solve (GSSSS) integrator function information:
% -------------------------------------------------------------------------
% function [u,ut,utt,Feff,kiter] = ...
% GSSSS(ExcData,Fint_K_C,m,AlgID,rinf,varargin)
% General linear or nonlinear explicit or implicit direct time
% integration of second order differential equations of SDOF or MDOF
% dynamic systems
%
% Description
% The General Single Step Single Solve (GSSSS) family of algorithms
% published by X.Zhou & K.K.Tamma (2004) is employed for direct time
% integration of the general linear or nonlinear structural Single
% Degree of Freedom (SDOF) or Multiple Degree of Freedom (MDOF) dynamic
% problem. Selection among 9 algorithms, all designed according to the
% above journal article, can be made in this routive. These algorithms
% encompass the scope of Linear Multi-Step (LMS) methods and are limited
% by the Dahlquist barrier theorem (Dahlquist,1963).
%
% Input parameters
% ExcData: matrix of two columns, the first column is time and the
% second is the imposed acceleration at the base.
% Fint_K_C: function handle which defines the force - displacement -
% velocity relation of the structure to be analysed. The definition of
% fun must be of the type: [Fint,K,C]=Fint_K_C(u,ut), where Fint is the
% internal force (sum of forces due to stiffness and damping) of the
% structure at displacement u and velocity ut, K and C are the tangent
% stiffness matrix and tangent damping matrix at the same displacement
% and velocity values. Type help Fint_K_C for details.
% m: mass matrix of the structure
% AlgID: ID of the algorithm to be used for the integration. It can be a
% row vector for commonly used algorithms or a suitable string for
% superior optimally designed algorithms. Type help GSSSS for more
% details
% rinf: Minimum absolute value of the eigenvalues of the amplification
% matrix. For the amplification matrix see eq.(61) in Zhou & Tamma
% (2004).
% varargin: optional arguments as follows:
% inflvec: influence vector. It determines the dofs at which the
% acceleration prescribed in ExcData will be imposed (column vector,
% number of Dofs-by-1). Default value 1 for SDOF and ones(DOFs,1)
% for MDOF.
% u0: initial displacement (column vector, number of Dofs-by-1).
% Default value 0 for SDOF and zeros(DOFs,1) for MDOF.
% ut0: initial velocity (column vector, number of Dofs-by-1)
% Default value 0 for SDOF and zeros(DOFs,1) for MDOF.
% maxtol: maximum tolerance of convergence of the Full Newton
% Raphson method for numerical computation of acceleration (scalar).
% Used only for implicit integration. Default value 0.1.
% kmax: maximum number of iterations per increment (scalar). If k=0
% then the integration is explicit and maxtol is not taken into
% account. If k>0 then the integration is implicit. Default value
% 10.
%
% Output parameters
% u: time-history of displacement
% ut: time-history of velocity
% utt: time-history of acceleration
% Feff: time-history of effective force (due to imposed acceleration,
% relative displacement and relative velocity)
% kiter: iterations per increment
%
% Copyright (c) 09-Dec-2013
% George Papazafeiropoulos
% First Lieutenant, Infrastructure Engineer, Hellenic Air Force
% Civil Engineer, M.Sc., Ph.D. candidate, NTUA
% Email: [e-mail]
% Website: http://users.ntua.gr/gpapazaf/
% _________________________________________________________________________