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Time Dependence
Published in Juan C. Heinrich, Darrell W. Pepper, Intermediate Finite Element Method, 2017
Juan C. Heinrich, Darrell W. Pepper
We must now address the solution of the system of ordinary differential equations given by Eq. (7.8). There are an abundance of methods available in the literature, particularly when the equations are stiff. Comprehensive studies have been provided by Richtmeyer and Morton (1963) and Lambert (1973). However, a majority of these methods were developed to integrate the equations obtained from finite difference approximations to the spatial derivatives. When finite element discretizations are used, the methods must be modified owing to the presence of the mass matrix. Here we will only establish the most basic methods available for the integration of Eq. (7.8) in time, which suffice in most general applications and can normally be easily adapted to more difficult situations requiring special treatment. For more comprehensive discussions and other methods not discussed here, the reader may consult the books by Zienkiewicz (1977), Bathe (1982), and Hughes (1987).
Finite Element Concepts In One-Dimensional Space
Published in Steven M. Lepi, Practical Guide to Finite Elements, 2020
Can the finite element method be used to solve problems associated with accelerating bodies? The answer is yes. In addition, the finite element method is often used for modal and forced vibration analysis. Fundamental to solving problems of dynamics using finite element analysis is the creation of a mass matrix, in addition to the finite element stiffness matrix. Finite element solutions to dynamics problems are beyond the scope of this text but are considered in Bathe [2] and Cook [17]; a very informative paper covering various aspects of dynamics problems is provided by Clough [18].
Initial boundary value problem
Published in Adnan Ibrahimbegovic, Naida Ademovicć, Nonlinear Dynamics of Structures Under Extreme Transient Loads, 2019
Adnan Ibrahimbegovic, Naida Ademovicć
The most costly phase in the proposed implementation of the central difference scheme clearly pertains to the solution of a set of algebraic equations for computing the acceleration vector an + 1. However, this cost can be reduced considerably by using a diagonal form of the mass matrix, which allows for the solution to be obtained with the number of operations equal only to the number of equations “n.” With this kind of computational efficiency, we can easily accept a very small time step that is often required to meet the conditional stability of the explicit scheme. For example, for the 1D hyperelastic bar (with free energy density ψ(λ)) and 2-node finite element approximations (with a typical element length le), the conditional stability of the central difference scheme requires (e.g. Oden and Fost 1973) the time step “h” no larger than: h≤le6cmax;cmax=max∀λ>0][C˜(λ)ρ1/2;C˜(λ)=d2ψ(λ)dλ2
A framework for rapid virtual prototyping: a case study with the Gunnerus research vessel
Published in Ship Technology Research, 2023
Pierre Major, Rami Zghyer, Houxiang Zhang, Hans Petter Hildre
The mass matrix elements are calculated from rigid body mass and inertia. Forced oscillations of the vessel generate outgoing waves and oscillating pressure field on the hull. Integration of this pressure over the wetted surface gives estimates for added mass forces and damping forces proportional to body acceleration and velocity, respectively. The restoring hydrostatic force behaves like a spring force and is a function of hull geometry, water density, gravitational acceleration, and mass distribution; hence, they are independent of waves or ship speed. ShipX estimates restoring coefficients in heave, roll, pitch, and coupled heave-roll degrees of freedom.