An Efficient Newton Barrier Method for Minimizing a Sum of Euclidean Norms

Andersen, Knud D.

doi:10.1137/0806006

Cited by 41 publications

(26 citation statements)

References 19 publications

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“…While linear programming involves piecewise linear yield functions, nonlinear programming involves nonlinear yield surfaces. Much progress has been made in developing numerical procedures for limit analysis problems [5][6][7][8][9][10]. Current research is focussing on developing limit analysis tools which are sufficiently efficient and robust to be of use to engineers working in practice.…”

Section: Introductionmentioning

confidence: 99%

Limit analysis of plates using the EFG method and second‐order cone programming

Gilbert

Askes

2009

Numerical Meth Engineering

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Published paperLe, Canh V., Gilbert, Matthew and Askes, Harm (2009) SUMMARYThe meshless Element-Free Galerkin (EFG) method is extended to allow computation of the limit load of plates. A kinematic formulation which involves approximating the displacement/velocity field using the moving least squares technique is developed. Only one displacement variable is required for each EFG node, ensuring that the total number of variables in the resulting optimization problem is kept to a minimum, with far fewer variables being required compared with finite element formulations. A stabilized conforming nodal integration scheme is extended to plastic plate bending problems. The evaluation of integrals at nodal points using curvature smoothing stabilization both keeps the size of the optimization problem small and also results in stable and accurate solutions. Difficulties imposing essential boundary conditions are overcome by enforcing directly displacements at the nodes. The formulation can be expressed as the problem of minimizing a sum of Euclidean norms subject to a set of equality constraints. This non-smooth minimization problem can be transformed into a form suitable for solution using Second-Order Cone Programming (SOCP). The procedure is applied to several benchmark problems and is found in practice to generate good upper bound solutions for benchmark problems.

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Section: Introductionmentioning

confidence: 99%

Limit analysis of plates using the EFG method and second‐order cone programming

Gilbert

Askes

2009

Numerical Meth Engineering

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“…To deal with this case, one can use, for instance, interior point methods [Luksan et al 2007;Schittkowski 2008;Gould et al 2003]. For simplicity, we instead employ a smooth approximation of the 1 -norm introduced in [El-Attar et al 1979] (hyperboloid approximation [Andersen 1996]). start from a relative large ε and decrease it gradually throughout our minimization procedure.…”

Section: Smooth Approximation Of 1 Normmentioning

confidence: 99%

“…Our formulation of the necessary condition for polycube also utilizes the sparsity property of the 1 norm, albeit in a very different geometric context. Because the 1 norm of the normal is non-linear in the node position, we adopt existing numerical techniques [El-Attar et al 1979;Andersen 1996] to turn our 1 minimization into a smooth optimization problem.…”

Section: Related Workmentioning

confidence: 99%

ℓ ₁ -Based Construction of Polycube Maps from Complex Shapes

et al. 2014

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Polycube maps of triangle meshes have proved useful in a wide range of applications including texture mapping and hexahedral mesh generation. However, constructing either fully automatically or with limited user control a low-distortion polycube from a detailed surface remains challenging in practice. We propose a variational method for deforming an input triangle mesh into a polycube shape through minimization of the 1 -norm of the mesh normals, regularized via an as-rigid-as-possible volumetric distortion energy. Unlike previous work, our approach makes no assumption on the orientation, or on the presence of features in the input model. User-guided control over the resulting polycube map is also offered to increase design flexibility. We demonstrate the robustness, efficiency and controllability of our method on a variety of examples, and explore applications in hexahedral remeshing and quadrangulation.

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“…This algorithm involves 2 objective functions, the so-called primal (P) and dual (D). The primal function takes greater values than the dual over all feasible points of the dual variables, except for a single point on which the 2 functions take the same value [11,[17][18][19].…”

Section: Introductionmentioning

confidence: 99%

“…Addressing the problem of minimizing the sum of Euclidean norms based on a technique, the so-called primal-dual interior-point method (PD-IPM) provides a new class of methods for TV regularization [11][12].…”

Section: Introductionmentioning

confidence: 99%

Improving the performance of primal--dual interior-point method in inverse conductivity problems

Javaherian¹,

Amir²,

Faghihi³

et al. 2015

Turk J Elec Eng & Comp Sci

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This study improves the performance of primal-dual interior-point method in inverse conductivity problems via replacing the conventional, complicatedly calculated scalar regularization parameter with a diagonal matrix termed "multi-regularization parameter matrix" here. The solution of the PD-IPM depends considerably on the choice of the regularization parameter. Calculation of the optimal regularization parameter, which yields the most accurate solution, is not simple due to the long iterative nature of the algorithm. The objective optimization, which is implemented by minimizing error in the solutions over an extensive range of the regularization parameters, yields the most accurate solution that can be achieved, although this method is not applicable in reality due to lack of knowledge about the actual conductivity field. However, the modified algorithm not only solves the problem independently using the regularization parameter, but also increases the accuracy of the solution, as well as its sharpness in comparison to the objective optimization.

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An Efficient Newton Barrier Method for Minimizing a Sum of Euclidean Norms

Cited by 41 publications

References 19 publications

Limit analysis of plates using the EFG method and second‐order cone programming

Limit analysis of plates using the EFG method and second‐order cone programming

ℓ ₁ -Based Construction of Polycube Maps from Complex Shapes

Improving the performance of primal--dual interior-point method in inverse conductivity problems

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An Efficient Newton Barrier Method for Minimizing a Sum of Euclidean Norms

Cited by 41 publications

References 19 publications

Limit analysis of plates using the EFG method and second‐order cone programming

Limit analysis of plates using the EFG method and second‐order cone programming

ℓ 1 -Based Construction of Polycube Maps from Complex Shapes

Improving the performance of primal--dual interior-point method in inverse conductivity problems

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Product

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ℓ ₁ -Based Construction of Polycube Maps from Complex Shapes