Analytical Shape Derivatives of the MFIE System Matrix Discretized With RWG Functions

The efficient and easy computational implementation of multibody dynamic formulations is an important issue. In this paper, a parametric generalized coordinate formulation is proposed as a new approach to formulate joint constraint equations. By introducing the parametric generalized coordinates in the constraint equations, the complexity of the equations is significantly reduced. The number of arithmetic operations required to compute the derivatives of the constraints is drastically decreased depending on the type of joint involved (i.e. especially when the proposed method is applied to joints having lower degrees of freedom). Furthermore, the second derivatives of the constraint equations tend to have a large portion of zero sub-matrices, which simplifies the system matrix of the governing equations and allows for application of efficient sparse matrix techniques. Although the proposed approach has the drawback of a larger set of coordinates than a conventional Cartesian coordinate approach resulting in slightly less efficient computation, a systematic and easy formulation is achieved, which may relieve the implementation complexity for program developers. The dynamic analysis of a multibody slider-crank system is carried out to demonstrate the validity of the proposed formulation.

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A Fully Coupled Scheme for Viscous Flows in Regular and Irregular Domains Using Compact Integrated RBF Approximation

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.553.138 ◽

2014 ◽

Vol 553 ◽

pp. 138-143 ◽

Cited By ~ 1

Author(s):

C.M.T. Tien ◽

N. Thai-Quang ◽

N. Mai-Duy ◽

C. D. Tran ◽

T. Tran-Cong

Keyword(s):

Viscous Flows ◽

Navier Stokes ◽

Test Problems ◽

System Matrix ◽

Irregular Domains ◽

Navier Stokes Equation ◽

Present Scheme ◽

Pressure Fields ◽

Derivatives Of ◽

Fully Coupled

In this study, we present a numerical discretisation scheme, based on a fully coupled approach and compact local integrated radial basis function (CIRBF) approximations, to solve the Navier-Stokes equation in rectangular/non-rectangular domains. The velocity and pressure fields are simulated in a fully coupled manner [1] with Cartesian grids. The field variables are locally approximated in each direction by using CIRBF approximations defined over 3-node stencils, where nodal values of the first-and second-order derivatives of the field variables are also included [2, 3]. The present scheme, whose system matrix is sparse, is verified through the solutions of several test problems including Taylor-Green vortices. Highly accurate solutions are obtained.

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Passive Control of Singularities by Topological Optimization: The Second-Order Mixed Shape Derivatives of Energy Functionals for Variational Inequalities

Springer Optimization and Its Applications - Advances in Mathematical Modeling, Optimization and Optimal Control ◽

10.1007/978-3-319-30785-5_4 ◽

2016 ◽

pp. 65-102

Author(s):

Günter Leugering ◽

Jan Sokołowski ◽

Antoni Żochowski

Keyword(s):

Variational Inequalities ◽

Passive Control ◽

Second Order ◽

Topological Optimization ◽

Energy Functionals ◽

Shape Derivatives ◽

Derivatives Of

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Shape derivatives for an augmented Lagrangian formulation of elastic contact problems

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2020063 ◽

2020 ◽

Author(s):

Bastien Chaudet-Dumas ◽

Jean Deteix

Keyword(s):

Augmented Lagrangian ◽

Sufficient Conditions ◽

Contact Problems ◽

Lagrangian Formulation ◽

Directional Derivatives ◽

Practical Reasons ◽

Process Conditions ◽

Shape Derivatives ◽

Derivatives Of ◽

Augmented Lagrangian Formulation

This work deals with shape optimization of an elastic body in sliding contact (Signorini) with a rigid foundation. The mechanical problem is written under its augmented Lagrangian formulation, then solved using a classical iterative approach. For practical reasons we are interested in applying the optimization process with respect to an intermediate solution produced by the iterative method. Due to the projection operator involved at each iteration, the iterate solution is not classically shape differentiable. However, using an approach based on directional derivatives, we are able to prove that it is conically differentiable with respect to the shape, and express sufficient conditions for shape differentiability. Finally, from the analysis of the sequence of conical shape derivatives of the iterative process, conditions are established for the convergence to the conical derivative of the original contact problem.

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