Dynamic Analysis of Soft Tissues Using a State Space Model

2008 ◽  
Author(s):  
Assimina Pelegri ◽  
Baoxiang Shan
Keyword(s):  
Differential Equation ◽  
Dynamic Analysis ◽  
State Space ◽  
Soft Tissues ◽  
Order Differential Equation ◽  
Computational Cost ◽  
Second Order ◽  
P Element ◽  
Fe Model ◽  
Second Order Differential Equation

Research on the biomechanical behavior of soft tissues has drawn a lot of recent attention due to its application in tumor pathology, rehabilitation, surgery and biomaterial implants. In this study a finite element (FE) model is applied to represent soft tissues and phantoms with complex geometry and heterogeneous material properties. A solid 3D mixed u-p element S8P0 (8-node for displacement and 1-node for internal pressure) is implemented to capture the near-incompressibility inherent in soft tissues. A dynamic analysis of soft tissues’ response to excitation is explored in which, the second order differential equation representing the soft tissues in FE necessitates a time-consuming numerical solution procedure. Moreover, the second-order representation is complicated in estimating the tissue mechanical properties by inverse procedure. Thus, a state space (SS) model is used to equivalently represent soft tissues by transforming the second-order differential equation into a system of linear first-order differential equations. The linear and time-invariant SS representation of soft tissues for general dynamic analysis can reduce the computational cost and a provide framework for the “forward” simulation and “inverse” identification of soft tissues.

A mechanical method for the solution of second-order linear differential equations

1931 ◽  
Vol 27 (4) ◽  
pp. 546-552 ◽  
Author(s):  
E. C. Bullard ◽  
P. B. Moon
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Order Differential Equation ◽  
Second Order ◽  
Linear Differential Equations ◽  
Mechanical Method ◽  
Order Linear ◽  
Second Order Differential Equation ◽  
Linear Differential

A mechanical method of integrating a second-order differential equation, with any boundary conditions, is described and its applications are discussed.

scholarly journals Periodic solutions for periodic second-order differential equations with variable potentials

2018 ◽  
Vol 24 (2) ◽  
pp. 127-137
Author(s):  
Jaume Llibre ◽  
Ammar Makhlouf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Second Order Differential Equation ◽  
Existence Of Periodic Solutions

Abstract We provide sufficient conditions for the existence of periodic solutions of the second-order differential equation with variable potentials {-(px^{\prime})^{\prime}(t)-r(t)p(t)x^{\prime}(t)+q(t)x(t)=f(t,x(t))} , where the functions {p(t)>0} , {q(t)} , {r(t)} and {f(t,x)} are {\mathcal{C}^{2}} and T-periodic in the variable t.

Numerical methods for solving the first-kind boundary value problem for a linear second-order differential equation with a deviating argument

2017 ◽  
Vol 23 (2) ◽  
Author(s):  
Muhad H. Abregov ◽  
Vladimir Z. Kanchukoev ◽  
Maryana A. Shardanova
Keyword(s):  
Differential Equation ◽  
Numerical Methods ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Second Order ◽  
Deviating Argument ◽  
Second Order Differential Equation

AbstractThis work is devoted to the numerical methods for solving the first-kind boundary value problem for a linear second-order differential equation with a deviating argument in minor terms. The sufficient conditions of the one-valued solvability are established, and the a priori estimate of the solution is obtained. For the numerical solution, the problem studied is reduced to the equivalent boundary value problem for an ordinary linear differential equation of fourth order, for which the finite-difference scheme of second-order approximation was built. The convergence of this scheme to the exact solution is shown under certain conditions of the solvability of the initial problem. To solve the finite-difference problem, the method of five-point marching of schemes is used.

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