Tuned test problems for numerical methods in engineering

Sinclair, G. B.; Anaya-Dufresne, Manuel; Meda, G.; Okajima, Mitsuharu

doi:10.1002/(sici)1097-0207(19971130)40:22<4183::aid-nme252>3.0.co;2-7

Int. J. Numer. Meth. Engng.

1997

DOI: 10.1002/(sici)1097-0207(19971130)40:22<4183::aid-nme252>3.0.co;2-7

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Tuned test problems for numerical methods in engineering

G. B. Sinclair

Manuel Anaya-Dufresne

G. Meda

et al.

Abstract: When numerical methods are applied in engineering, verification is essential. Test problems can help in meeting this requirement. This is especially so when these problems are tuned to the application at hand. Unfortunately, in practice such tuned test problems are not always directly available. For these cases, this paper describes an approach for constructing tuned test problems. The approach may be used on ordinary or partial differential equations, be they linear or non-linear. This is demonstrated on a nu… Show more

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“…Solutions, however, were obtained in none of these attempts (under the same assumptions at which the classical solutions are constructed). As the exact solutions are of interest and can play an important role in verification of numerical codes [7], the search for such solutions, especially in the presence of friction stresses, is an urgent task.…”

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Compression of a plastic porous material between rotating plates

Alexandrov

Pirumov

Chesnikova

2009

J Appl Mech Tech Phy

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The problem of a plane flow of a rigid-plastic porous material between two rotating rough plates with no material flux through the point of their rotation and with a uniform distribution of porosity at the initial instant is considered under the assumption that the material behavior follows the cylindrical yield condition and the associated flow rule. The solution reduces to consecutive calculation of several ordinary integrals. It is demonstrated that the solution behavior depends on the angle between the plates, and the value of porosity at a certain stage of the deformation process can be equal to zero.In the classical theory of plasticity of incompressible materials, there are many analytical solutions obtained by an inverse method [1,2]. In the theory of plasticity of porous materials, such solutions seem to be lacking (even for the initial flow), except for a uniform stress-strain state and extremely simple problems with the friction stress neglected, for instance, problems of contraction of a hollow cylinder in a plane strain state (initial flow), contraction of a hollow spherical shell (initial flow), and flow through a convergent channel [3]. The classical problems of the plasticity theory [the most typical problem is the Prandtl problem of compression of a layer between rough plates (see, e.g., [1])], which have fairly simple solutions in the case of an incompressible rigid perfectly/plastic material (and their extensions to other models of incompressible materials [4]), cannot be extended to models of plastically compressible materials. In particular, attempts of problem extension were made in [5] for the Prandtl problem and in [6] for the flow through an infinitely convergent channel. Solutions, however, were obtained in none of these attempts (under the same assumptions at which the classical solutions are constructed). As the exact solutions are of interest and can play an important role in verification of numerical codes [7], the search for such solutions, especially in the presence of friction stresses, is an urgent task.We assume that the material behavior obeys the cylindrical yield condition proposed in [8] and written in the formHere σ eq is the equivalent stress, σ is the mean stress, τ s is the shear yield stress, and p s is the yield stress under hydrostatic compression. The equivalent and mean stresses are defined by the relations σ eq = 3/2 (τ ij τ ij ) 1/2 and σ = σ ij δ ij /3, where σ ij are the stress tensor components, τ ij are the stress deviator components, and δ ij is the Kronecker symbol. The quantities p s and τ s are assumed to be known functions of porosity η. As η → 0, we have p s → ∞, and quantity τ s tends to the shear yield stress of the base material. Though the yield condition (1) is extremely simple, it can describe some experimental observations [8]. Other solutions with the use of the yield condition (1) were obtained in [9,10]. Note that various flow regimes are possible if this condition is applied. Nevertheless, the regime described by the relations σ eq √ 3 τ s and...

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Compression of a plastic porous material between rotating plates

Alexandrov

Pirumov

Chesnikova

2009

J Appl Mech Tech Phy

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“…In addition, exact solutions are used for testing numerical algorithms (see, e.g., [9]). Within the framework of the gradient theories of plasticity, solutions of this class were obtained for twisting a cylindrical rod [1] and expansion of a hollow cylinder under plane strain conditions [8].…”

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Solving the problem of expansion and extension of a hollow cylinder with the use of the gradient plasticity theory

Alexandrov

Lyamina

2009

J Appl Mech Tech Phy

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An exact solution of the problem of expansion and extension of a hollow cylinder, which generalizes the known solutions based on the classical and gradient theories of plasticity under plane strain conditions, is obtained. The numerical solution reduces to calculating several ordinary integrals. The effect of the characteristic size of the material on the solution behavior near the cylinder orifice is studied.Gradient theories of plasticity are used to describe some effects arising near stress concentrators and during deformation of small-size bodies (see, e.g., [1][2][3][4][5][6][7][8]). In particular, Novopashin et al.[2] performed a detailed analytical analysis of gradient yield criteria and proposed their physical interpretation.As with other theories, analytical and semi-analytical solutions are used to study the general properties of gradient models. In addition, exact solutions are used for testing numerical algorithms (see, e.g., [9]). Within the framework of the gradient theories of plasticity, solutions of this class were obtained for twisting a cylindrical rod [1] and expansion of a hollow cylinder under plane strain conditions [8]. One more exact solution is obtained in the present paper.We use the gradient theory of plasticity proposed in [5], in which the yield condition has the form σ eq = σ 0 [f 2 (ε eq , ξ eq ) + l 0 |∇ε eq |] 1/2 .Here, σ eq = 3/2 (τ ij τ ij ) 1/2 is the equivalent stress, τ ij are the deviator components of the stress tensor, ξ eq = 2/3 (ξ ij ξ ij ) 1/2 is the equivalent strain rate, ξ ij are the components of the strain rate tensor, σ 0 = const, ε eq is the equivalent plastic strain defined by the equationand d/dt is the material derivative. The dependence of the function f included into Eq. (1) on the equivalent plastic strain and equivalent strain rate is usually taken in the same form as that used in the constitutive relations of the plasticity theory containing no gradient terms. In the present work, f is assumed to be independent of ξ eq , and the dependence of f on ε eq is taken in the form of Swift's law [10] f (ε eq ) = (1 + ε eq /ε 0 ) n , ε 0 = const, n = const.In Eq. (1), the gradient term is introduced with the use of the characteristic length l 0 , which is a material property, and the gradient of the equivalent plastic strain ∇ε eq . In addition to the plasticity condition (1), the constitutive equations include the associated plastic flow rule, which has the following form in the case considered:Ishlinskii Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow 119526; lyamina@inbox.ru.

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Exact analytical solution to a rigid/plastic problem with hardening and damage evolution

Alexandrov¹,

Goldstein²

1999

Comptes Rendus de l'Académie des Sciences - Series IIB - Mechan

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Tuned test problems for numerical methods in engineering

Cited by 5 publications

References 19 publications

Compression of a plastic porous material between rotating plates

Compression of a plastic porous material between rotating plates

Solving the problem of expansion and extension of a hollow cylinder with the use of the gradient plasticity theory

Exact analytical solution to a rigid/plastic problem with hardening and damage evolution

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