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A Newton Barrier method for Minimizing a Sum of Euclidean Norms subject to linear equality constraints
, 1995
"... An algorithm for minimizing a sum of Euclidean Norms subject to linear equality constraints is described. The algorithm is based on a recently developed Newton barrier method for the unconstrained minimization of a sum of Euclidean norms (MSN ). The linear equality constraints are handled using an e ..."
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Cited by 27 (2 self)
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An algorithm for minimizing a sum of Euclidean Norms subject to linear equality constraints is described. The algorithm is based on a recently developed Newton barrier method for the unconstrained minimization of a sum of Euclidean norms (MSN ). The linear equality constraints are handled using an exact L 1 penalty function which is made smooth in the same way as the Euclidean norms. It is shown that the dual problem is to maximize a linear objective function subject to homogeneous linear equality constraints and quadratic inequalities. Hence the suggested method also solves such problems efficiently. In fact such a problem from plastic collapse analysis motivated this work. Numerical results are presented for large sparse problems, demonstrating the extreme efficiency of the method. Keywords: Sum of Norms, Nonsmooth Optimization, Duality, Newton Barrier Method. AMS(MOS) subject classification: 65K05, 90C06, 90C25, 90C90. Abbreviated title: A Newton barrier method. Supported by the ...
Computing Limit Loads By Minimizing a Sum of Norms
 IFIP
, 1994
"... This paper treats the problem of computing the collapse state in limit analysis for a solid with a quadratic yield condition, such as, for example, the Mises condition. After discretization with the finite element method, using divergencefree elements for the plastic flow, the kinematic formulation ..."
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Cited by 21 (3 self)
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This paper treats the problem of computing the collapse state in limit analysis for a solid with a quadratic yield condition, such as, for example, the Mises condition. After discretization with the finite element method, using divergencefree elements for the plastic flow, the kinematic formulation turns into the problem of minimizing a sum of Euclidean vector norms, subject to a single linear constraint. This is a nonsmooth minimization problem, since many of the norms in the sum may vanish at the optimal point. However, efficient solution algorithms for this particular convex optimization problem have recently been developed. The method is applied to test problems in limit analysis in two different plane models: plane strain and plates. In the first case more than 80 percent of the terms in the sum are zero in the optimal solution, causing severe illconditioning. In the last case all terms are nonzero. In both cases the algorithm works very well, and problems are solved which are l...
Mesh adaptive computation of upper and lower bounds in limit analysis
"... An efficient procedure to compute strict upper and lower bounds for the exact collapse multiplier in limit analysis is presented, with a formulation that explicitly considers the exact convex yield condition. The approach consists of two main steps. First, the continuous problem, under the form of t ..."
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Cited by 6 (1 self)
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An efficient procedure to compute strict upper and lower bounds for the exact collapse multiplier in limit analysis is presented, with a formulation that explicitly considers the exact convex yield condition. The approach consists of two main steps. First, the continuous problem, under the form of the static principle of limit analysis, is discretized twice (one per bound) using particularly chosen finite element spaces for the stresses and velocities that guarantee the attainment of an upper or a lower bound. The second step consists of solving the resulting discrete nonlinear optimization problems. These are reformulated as secondorder cone programs, which allows for the use of primal–dual interior point methods that optimally exploit the convexity and duality properties of the limit analysis model. To benefit from the fact that collapse mechanisms are typically highly localized, a novel method for adaptive meshing is introduced. The method first decomposes the total bound gap as the sum of positive contributions from each element in the mesh and then refines those elements with higher contributions. The efficiency of the methodology is illustrated with applications in plane stress and plane strain problems. Copyright q 2008 John Wiley
pp. X–XX NUMERICAL SOLUTION OF A VARIATIONAL PROBLEM ARISING IN STRESS ANALYSIS: THE VECTOR CASE
"... (Communicated by the associate editor name) Dedicated to Professor Roger Temam on the occasion of his 70th birthday Abstract. In this article, we discuss the numerical solution of a constrained minimization problem arising from the stress analysis of elastoplastic bodies. This minimization problem ..."
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(Communicated by the associate editor name) Dedicated to Professor Roger Temam on the occasion of his 70th birthday Abstract. In this article, we discuss the numerical solution of a constrained minimization problem arising from the stress analysis of elastoplastic bodies. This minimization problem has the flavor of a generalized nonsmooth eigenvalue problem, with the smallest eigenvalue corresponding to the load capacity ratio of the elastic body under consideration. An augmented Lagrangian method, together with finite element approximations, is proposed for the computation of the optimum of the nonsmooth objective function, and the corresponding minimizer. The augmented Lagrangian approach allows the decoupling of some of the nonlinearities and of the differential operators. Similarly an appropriate Lagrangian functional, and associated Uzawa algorithm with projection, are introduced to treat nonsmooth equality constraints. Numerical results validate the proposed methodology for various twodimensional geometries. 1. Introduction and Motivations
9 th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability PMC2004 COMPUTATION OF UPPER AND LOWER BOUNDS IN LIMIT ANALYSIS USING SECONDORDER CONE PROGRAMMING AND MESH ADAPTIVITY
"... An efficient procedure to compute strict upper and lower bounds for the exact collapse multiplier in limit analysis is presented. The approach consists of two main steps. First, the continuous problem, under the form of the static principle of limit analysis, is discretized twice (one per bound) usi ..."
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An efficient procedure to compute strict upper and lower bounds for the exact collapse multiplier in limit analysis is presented. The approach consists of two main steps. First, the continuous problem, under the form of the static principle of limit analysis, is discretized twice (one per bound) using particularly chosen finite element spaces for the stresses and velocities that guarantee the attainment of an upper or a lower bound. The second step consists of solving the resulting discrete nonlinear optimization problems. Towards this end, they are reformulated as secondorder cone programs, which allows for the use of primaldual interior point methods that optimally exploit the convexity and duality properties of the limit analysis model. To benefit from the fact that collapse mechanisms are typically highly localized, a novel method for adaptive meshing is introduced based on decomposing the total bound gap as the sum of positive elemental contributions from each element in the mesh. Additionally, standalone computational certificates that allow the bounds to be verified independently, without recourse to the original computer program, are also provided. The efficiency of the methodology is illustrated with applications in plane stress and plane strain, demonstrating that it can be used in complex, realistic problems.