Related papers: An elementary formula for computing shape derivati…
In industry, shape optimization problems are of utter importance when designing structures such as aircraft, automobiles and turbines. For many of these applications, the structure changes over time, with a prescribed or non-prescribed…
In this paper we study an abstract framework for computing shape derivatives of functionals subject to PDE constraints. We revisit the Lagrangian approach using the implicit function theorem in an abstract setting tailored for applications…
We present a simple formalism for the calculation of the derivatives of the electronic density matrix at any order, within density functional theory. Our approach, contrary to previous ones, is not based on the perturbative expansion of the…
The Riemann-Liouville formula for fractional derivatives and integrals (differintegration) is used to derive formulae for matrix order derivatives and integrals. That is, the parameter for integration and differentiation is allowed to…
A procedure to obtain differentiation matrices is extended straightforwardly to yield new differentiation matrices useful to obtain derivatives of complex rational functions. Such matrices can be used to obtain numerical solutions of some…
This paper introduces a new computational framework to derive electromagnetic field derivatives with respect to multiple design parameters up to any order with the Finite-Difference Time-Domain (FDTD) technique. Specifically, only one FDTD…
Starting from space-discretisation of Maxwell's equations, various classical formulations are proposed for the simulation of electromagnetic fields. They differ in the phenomena considered as well as in the variables chosen for…
We discuss automating the calculation of weak shape derivatives in the Unified Form Language (Aln{\ae}s et al., ACM Trans. Math. Softw., 2014) by introducing an appropriate additional step in the pullback from physical to reference space…
This paper introduces a new approach for the computation of electromagnetic field derivatives, up to any order, with respect to the material and geometric parameters of a given geometry, in a single Finite-Difference Time-Domain (FDTD)…
The simple product formulae for derivatives of scalar functions raised to different powers are generalized for functions which take values in the set of symmetric positive definite matrices. These formulae are fundamental in derivation of…
A natural consequence of the fractional calculus is its extension to a matrix order of differentiation and integration. A matrix-order derivative definition and a matrix-order integration arise from the generalization of the gamma function…
We introduce new results about the shape derivatives of scalar- and vector-valued functions, extending the results from (Dogan-Nochetto 2012) to more general surface energies. They consider surface energies defined as integrals over…
This paper is a generalization of the previous work (Yang et.al, J. Comput. Phys. 330 (2017), 863-883) to the 3-D irregular convex domains. The analytical calculation formula of fractional derivatives of finite element basis functions are…
In this paper we present a framework for automated shape differentiation in the finite element software NGSolve. Our approach combines the mathematical Lagrangian approach for differentiating PDE constrained shape functions with the…
In this work we deduce explicit formulae for the elements of the matrices that represent the action of integro-differential operators over the coefficients of generalized Fourier series. Our formulae are obtained by performing operations on…
We introduce a novel mesh-free and direct method for computing the shape derivative in PDE-constrained shape optimization problems. Our approach is based on a probabilistic representation of the shape derivative and is applicable for…
Scientific studies often require the precise calculation of derivatives. In many cases an analytical calculation is not feasible and one resorts to evaluating derivatives numerically. These are error-prone, especially for higher-order…
We establish derivative estimates of solution of elliptic system in narrow regions.
This short note provides an explicit description of the Fr\'echet derivatives of the principal square root matrix functional at any order. We present an original formulation that allows to compute sequentially the Fr\'echet derivatives of…
We use a basic martingale method to show a differentiation formula for the derivatives $$d(P_tf)(x_0)(v_0)={1\over t} E f(x_t) \int_0^t \langle Y(x_s)(v_s),dB_t\rangle_{R^m}.$$ These are proved first on $R^n$, then on manifolds. Afterwards…