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Thanks in advance for your time. Skip to content. Search for books, journals or webpages All Pages Books Journals. View on ScienceDirect. Editors: S. Imprint: Pergamon. Published Date: 1st January The first difficulty relates to the proper use of initial conditions resulting from the restrictions on the function variations. For a typical dynamic problem, one does not know how the considered system evolves at the end of the time interval. Second difficulty relates to the inability to incorporate irreversible phenomena.
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Somewhat earlier, Gurtin ; Gurtin a ,[ b ] introduced the convolution functional, and could reduce the initial value problem to an equivalent boundary value problem. However, the functional by Gurtin is complicated and it never can recover the original strong form.
Following the ideas of Tonti and Gurtin, Oden and Reddy Oden and Reddy extended the formulation to a large class of initial boundary problems in mechanics, especially for Hellinger-Reissner type mixed principles. More recently, Riewer Riewe ; Riewe adopted the use of fractional calculus to accommodate dissipative dynamical systems.
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This is an attractive idea, and many other researches including Agrawal Agrawal ; Agrawal ; Agrawal , Atanackovic et al. However, surprisingly, none of these papers include an analytical description validating their approach for the most fundamental case, a classical Kelvin-Voigt single-degree-of-freedom SDOF damped oscillator. With MCAP, a single scalar functional action provides the governing differential equations, along with all the pertinent boundary and initial conditions for conservative and non-conservative linear systems.
While EHP can be numerically implemented for viscoplasticity continuum dynamics, MCAP currently suffers from finding the explicit functional action for that problem. Since both methods provide sound basis to develop various space-time finite element methods for linear initial boundary value problems, here, the focus is initially on investigating their potential when employing higher-order temporal approximations. The remainder of the paper is organized as follows.
In Section Numerical implementation, discretization scheme and numerical algorithms are provided when temporally higher-order approximations are adopted in both approaches. Basic numerical properties of the developed methods are closely examined in Section Basic numerical properties. Then, some numerical examples are presented to investigate and to validate all of these developed algorithms for practical problems of the forced vibration in Section Numerical examples. Finally, some conclusions are provided in Section Conclusions. Then, the functional action A for the fixed time interval from t 0 to t is given by.
Such added terms have effect on confining a dynamical system to evolve uniquely from start to end with the unspecified values at the ends of the time interval such as and.
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Then, interpreting the unspecified initial terms as sequentially assigning the known initial values completes this formulation. Thus, in EHP, the given initial velocity is assigned first, and the given initial displacement u 0 is assigned next by. The subsequent zero-valued term 6 needs not appear explicitly in the new action variation, so that the new definition 4 with the sequential assigning process such as 5 and 6 can properly account for the initial value problems. It should be noted that in EHP, the dependent initial condition J 0 can be identified by. In Eq. To check this, let us substitute Eqs.
Then, we have. Doing integration by parts on and in Eq. With the underlined terms in Eq. Thus, in EHP, the action variation for this model is defined by adding up Eq. With Eq. Referred to Oldham and Spanier ; Samko et al. Meanwhile, the last term in Eq. After performing classical and fractional integration by parts on the appropriate terms in Eq.
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For arbitrary variations of u and J , Eq. In other words, in MCAP, we can identify the dependent initial conditions such as J 0 and from the usual given initial conditions u 0 and as well as the known initial impulse. As shown in Eq. However, MCAP still requires a generalized framework to embrace various irreversible phenomena. Also, it should be noted that any pair of complementary order of fractional derivatives in Eq. Consequently, we have C 0 temporal continuity requirement on primary variables and the variations, thus, there are many cases to develop higher order temporal finite element methods.
As we shall see in this section, three kinds of quadratic temporal finite element methods in each framework are developed, since they are practically sufficient and accurate in computational aspects as discussed next. Developed quadratic temporal finite element methods in each framework.
Also, represents linear momentum, where. For a representative case, Jquad algorithm can be obtained from the main approximations as. Substituting Eqs. While deriving Eqs.
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From either Eq. Previously, MCAP was numerically implemented through linear temporal shape functions for classical SDOF oscillators and systems that utilize fractional-derivative constitutive models by Dargush Here, continuing through this line, but, the quadratic temporal finite element methods are developed. As well described in Dargush , for any non-negative integer m and n , we have the following relation.
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To evaluate the convolution of semi-derivatives of polynomial shape functions, here, Eq. With approximations 58 - 67 , the convolution component in Eq. Each component of matrix in Eq. For a representative one, is computed as. With evaluation of typical integer order convolution components in Eq. Thus, the weak form reduces to the following:. Again, with the adoption of the same strategy as Eqs. For the undamped case with no external forcing conservative harmonic oscillator , Eqs.
In each Table, X and Y are given respectively in Eq. While deriving Eq.
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The stability and dissipative character of each developed method can be determined by considering the eigenvalues of A in Eq. Notice that the magnitude of all the eigenvalues including complex conjugate pairs in each Table is exactly equal to 1, which can be written simply as. Consequently, in addition to being time reversible, all the presented quadratic temporal finite element algorithms are also symplectic, energy conserving, and unconditionally stable for the undamped case. To check the period elongation property in each developed method, the method by Bathe ; Bathe and Wilson is used for free vibration of the undamped oscillator, where the ratio of the time-step h to the natural period T n is a control parameter.
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Figure 1. Full size image. Figure 2. SDOF elasto-viscoplastic model.
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