PurposeThe object of the paper is to illustrate how to obtain the topological derivative as a semidifferential in a general and practical mathematical setting for d-dimensional perturbations of a ...bounded open domain in the n-dimensional Euclidean space.Design/methodology/approachThe underlying methodology uses mathematical notions and powerful tools with ready to check assumptions and ready to use formulas via theorems on the one-sided derivative of parametrized minima and minimax.FindingsThe theory and the examples indicate that the methodology applies to a wide range of problems: (1) compliance and (2) state constrained objective functions where the coupled state/adjoint state equations appear without a posteriori substitution of the adjoint state.Research limitations/implicationsDirect approach that considerably simplifies the analysis and computations.Originality/valueIt was known that the shape derivative was a differential. But the topological derivative is only a semidifferential, that is, a one-sided directional derivative, which is not linear with respect to the direction, and the directions are d-dimensional bounded measures.
In this paper, the efficiency in mesh updating (r-adaptivity) of the Transfinite Mean value Interpolation (TMI) and its generalization (k-TMI) are compared on three standardized problems to the ...well-known Inverse Distance Weighted interpolation (IDW) and Radial Basis Function interpolation (RBF) for unstructured data points and the new k-Transfinite Barycentric Interpolation (k-TBI) for structured data points such as, for instance, curves or surfaces in 3D. This is achieved by introducing a dynamical version of these interpolations via an ordinary differential equation that can be solved by standard ODE methods that are more economical than, for instance, solving vector partial differential equations as in the pseudo-solid method.
A review of the very recent mathematical foundations of the k-TMI and k-TBI constructed from the function alone (standard) or from the function and its derivatives (enhanced) is provided in the first part of the paper.
This paper presents a new monolithic formulation targeted at the simulation of fluid-structure interaction. It combines the dynamic version of the recent k-TMI extension of the Transfinite Mean Value ...Interpolation (TMI) of Dyken and Floater with the ALE formulation of the Navier-Stokes equations for moving finite element meshes along with the Lagrange multiplier method to accurately predict parietal shear. This new integrated k-TMI/ALE formulation is stable for the extreme case of zero mass ratios.
In order to achieve prescribed drug release kinetics some authors have been investigating bi-phasic and possibly multi-phasic releases from blends of biodegradable polymers. Recently, experimental ...data for the release of paclitaxel have been published by Lao et al. (Lao and Venkatraman in J. Control. Release 130:9–14,
2008
; Lao et al. in Eur. J. Pharm. Biopharm. 70:796–803,
2008
). In Blanchet et al. (SIAM J. Appl. Math. 71(6):2269–2286,
2011
) we validated a two-parameter quadratic ordinary differential equation (ODE) model against their experimental data from three representative neat polymers. In this paper we provide a gradient flow interpretation of the ODE model. A three-dimensional partial differential equation (PDE) model for the drug release in their experimental set up is introduced and its parameters are related to the ones of the ODE model. The gradient flow interpretation is extended to the study of the asymptotic concentrations that are solutions of the PDE model to determine the range of parameters that are suitable to simulate complete or partial drug release.
Zhang SIAM J. Control Optim., 43 (2005), pp. 2157-2165 recently established the equivalence between the finiteness of the open loop value of a two-player zero-sum linear quadratic (LQ) game and the ...finiteness of its open loop lower and upper values. In this paper we complete and sharpen the results of Zhang for the finiteness of the lower value of the game by providing a set of necessary and sufficient conditions that emphasizes the feasibility condition: $(0,0)$ is a solution of the open loop lower value of the game for the zero initial state. Then we show that, under the assumption of an open loop saddle point in the time horizon $0,T$ for all initial states, there is an open loop saddle point in the time horizon $s,T$ for all initial times $s$, $0\le s<T$, and all initial states at time $s$. From this we get an optimality principle and adapt the invariant embedding approach to construct the decoupling symmetrical matrix function $P(s)$ and show that it is an $H^1(0,T)$ solution of the matrix Riccati differential equation. Thence an open loop saddle point in $0,T$ yields closed loop optimal strategies for both players. Furthermore, a necessary and sufficient set of conditions for the existence of an open loop saddle point in $0,T$ for all initial states is the convexity-concavity of the utility function and the existence of an $H^1(0,T)$ symmetrical solution to the matrix Riccati differential equation. As an illustration of the cases where the open loop lower/upper value of the game is $-\infty$/$+\infty$, we work out two informative examples of solutions to the Riccati differential equation with and without blow-up time.
Celotno besedilo
Dostopno za:
CEKLJ, DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
The object of this paper is to revisit the results of Bernhard J. Optim. Theory Appl., 27 (1979), pp. 51-69 on two-person zero-sum linear quadratic differential games and generalize them to utility ...functions without positivity assumptions on the matrices acting on the state variable and to linear dynamics with bounded measurable data matrices. Our paper specializes to state feedback via Lebesgue measurable affine closed loop strategies with possible non-$L^2$-integrable singularities. After sharpening the recent results of Delfour SIAM J. Control Optim., 46 (2007), pp. 750-774 on the characterization of the open loop lower and upper values of the game, it first deals with $L^2$-integrable closed loop strategies and then with the larger family of strategies that may have non-$L^2$-integrable singularities. A new conceptually meaningful and mathematically precise definition of a closed loop saddle point is introduced to simultaneously handle state feedbacks of the $L^2$ type and smooth locally bounded ones, except at most in the neighborhood of finitely many instants of time. A necessary and sufficient condition is that the free end problem be normalizable almost everywhere. This relaxation of the classical notion allows singularities in the feedback law at an infinite number of instants, including accumulation points that are not isolated. A complete classification of closed loop saddle points is given in terms of the convexity/concavity properties of the utility function, and connections are given with the open loop lower value, upper value, and value of the game.
Celotno besedilo
Dostopno za:
CEKLJ, DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
In order to achieve prescribed drug release kinetics some authors have been investigating biphasic and possibly multiphasic releases from blends of biodegradable polymers. Recently, experimental data ...for the release of paclitaxel have been published by Lao and Venkatraman J. Control. Release, 130 (2008), pp. 9-14 and Lao, Venkatraman, and Peppas Eur. J. Pharm. Biopharm., 70 (2008), pp. 796-803. The present paper validates three mathematical ordinary differential equation models against their experimental data: a quadratic one for the release of paclitaxel, a combination of this model with the idea of partition coefficients of Lao, Venkatraman, and Peppas Eur. J. Pharm. Biopharm., 70 (2008), pp. 796-803 for polymer blends, and the Verlhurst population model with a log kill law coupled with the drug release model that describes the evolution of smooth muscle cells in the presence of paclitaxel released from three neat polymers.
The notion of dose that comes from the biologists has been introduced by Delfour et al. (2005 SIAM J. Appl. Math. 65(3):858-881) in the context of coated stents to control restenosis. Assuming a ...stationary velocity profile of the blood flow in the lumen, it leads to a time-independent equation for the dose that considerably simplifies the analysis and the design problem. Under stable conditions the blood flow is pulsative, that is the velocity field can be assumed to be periodic. So it is necessary to justify the replacement of the periodic field by its time average over the pulsation period. In this paper, firstly we introduce the new unfolded dose and its equations without a priori constraint on the size of the period. So it can be used in biochemical problems where the period is large compared to the time constants of the system. Secondly, we show that, as the period goes to zero, the velocity field can be replaced by its average over the period. Numerical tests on a one-dimensional example are included to illustrate the theory.
Stents are used in interventional cardiology to keep a diseased vessel open. New stents are coated with a medicinal agent to prevent early reclosure due to the proliferation of smooth muscle cells. ...It is recognized that it is the dose of the agent that effectively controls the cells in the wall of the vessel. This paper focuses on the effect of the number of struts and the ratio between the coated area of the struts and the targeted area of the vessel on the design problem under set therapeutic bounds on the dose. It introduces mathematical models of the dose for a zero-thickness periodic stent and an asymptotic stent that will play a central role in our analysis. Theoretical and numerical results are presented along with their impact on the design process.
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