## General Single Step Single Solve integration algorithm

% 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: gpapazafeiropoulos@yahoo.gr

% Website: http://users.ntua.gr/gpapazaf/

% _________________________________________________________________________

### Cite As

George Papazafeiropoulos (2023). General Single Step Single Solve integration algorithm (https://www.mathworks.com/matlabcentral/fileexchange/44649-general-single-step-single-solve-integration-algorithm), MATLAB Central File Exchange. Retrieved .

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