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Biorthogonality Condition

The Experts below are selected from a list of 99 Experts worldwide ranked by ideXlab platform

S A Khuri – 1st expert on this subject based on the ideXlab platform

  • Biorthogonality Condition for creeping flow in wedge shaped trenches
    Applied Mathematics and Computation, 2003
    Co-Authors: S A Khuri

    Abstract:

    In the present paper, we derive the Biorthogonality Condition for Stokes flow in wedge-shaped trenches bounded by radial lines and concentric circles centered at the vertex of the wedge. This Condition, is a property satisfied by the eigenfunctions and adjoint eigenfunctions needed to compute the coefficients of the eigenfunction expansion solution of the corresponding Stokes flow problem.

  • Biorthogonality Condition for axisymmetric stokes flow in spherical geometries
    International Journal of Mathematics and Mathematical Sciences, 2000
    Co-Authors: S A Khuri

    Abstract:

    We derive the Biorthogonality Condition for axisymmetric Stokes flow in a re- gion between two concentric spheres. This Biorthogonality Condition is a property satis- fied by the eigenfunctions and adjoint eigenfunctions, which is needed to compute the coefficients of the eigenfunction expansion solution of the corresponding creeping flow problem.

  • Biorthogonality Condition for creeping motion in annular trenches
    International Journal of Mathematics and Mathematical Sciences, 2000
    Co-Authors: S A Khuri

    Abstract:

    The Biorthogonality Condition for Stokes flow in annular trenches
    bounded by horizontal parallel planes and concentric vertical
    cylinders is derived. This Condition, is needed to compute the
    coefficients of the eigenfunction expansion solution of the
    corresponding Stokes flow problem.

Stefan Hartmann – 2nd expert on this subject based on the ideXlab platform

  • Unilateral non‐linear dynamic contact of thin‐walled structures using a primal‐dual active set strategy
    International Journal for Numerical Methods in Engineering, 2020
    Co-Authors: Stefan Hartmann, Ekkehard Ramm, Stephan Brunssen, Barbara Wohlmuth

    Abstract:

    The efficient modelling of three-dimensional contact problems is still a challenge in non-linear implicit structural analysis. We use a primal-dual active set strategy (SIAM J. Optim. 2003; 13:865-888), based on dual Lagrange multipliers (SIAM J. Numer. Anal. 2000; 38:989-1012) to handle the non-linearity of the contact Conditions. This allows us to enforce the contact constraints in a weak, integral sense without any additional parameter. Due to the Biorthogonality Condition of the basis functions, the Lagrange multipliers can be locally eliminated. We perform a static condensation to achieve a reduced system for the displacements. The Lagrange multipliers, representing the contact pressure, can be easily recovered from the displacements in a variationally consistent way. For the application to thin-walled structures we adapt a three-dimensional non-linear shell formulation including the thickness stretch of the shell to contact problems. A reparametrization of the geometric description of the shell body gives us a surface-oriented shell element, which allows the application of contact Conditions directly to nodes lying on the contact surface. Shell typical locking phenomena are treated with the enhanced-assumed-strain-method and the assumed-natural-strain-method. The discretization in time is done with the implicit Generalized-α method (J. Appl. Mech. 1993; 60:371-375) and the Generalized Energy-Momentum Method (Comp. Methods Appl. Mech. Eng. 1999; 178:343-366) to compare the development of energies within a frictionless contact description. In order to conserve the total energy within the discretized frictionless contact framework, we follow an approach from Laursen and Love (Int. J. Numer. Methods Eng. 2002; 53:245-274), who introduced a discrete contact velocity to update the velocity field in a post-processing step. Various examples show the good performance of the primal-dual active set strategy applied to the implicit dynamic analysis of thin-walled structures.

  • A mortar based contact formulation for non-linear dynamics using dual Lagrange multipliers
    Finite Elements in Analysis and Design, 2008
    Co-Authors: Stefan Hartmann, Ekkehard Ramm

    Abstract:

    Many existing algorithms for the analysis of large deformation contact problems use a so-called node-to-segment approach to discretize the contact interface between dissimilar meshes. It is well known, that this discretization strategy may lead to problems like loss of convergence or jumps in the contact forces. Additionally it is popular to use penalty methods to satisfy the contact constraints. This necessitates a user defined penalty parameter the choice of which is somehow arbitrary, problem dependent and might influence the accuracy of the analysis. In this work, a frictionless segment-to-segment contact formulation is presented that does not require any user defined parameter to handle the non-linearity of the contact Conditions. The approach is based on the mortar method enforcing the compatibility Condition along the contact interface in a weak integral sense. The application of dual spaces for the interpolation of the Lagrange multiplier allows for a nodal decoupling of the contact constraints. A local basis transformation in combination with a primal-dual active set strategy enables the exact enforcement of the contact constraints via prescribed incremental boundary Conditions. Due to the Biorthogonality Condition of the basis functions the Lagrange multipliers can be locally eliminated. A static condensation leads to a reduced system of equations to be solved solely for the unknown nodal displacements. Thus the size of the system of equations remains constant during the whole calculation. The discrete Lagrange multipliers, representing the contact pressure, can be easily recovered from the displacements in a variationally consistent way. For the analysis of dynamic contact problems the proposed contact description is combined with the implicit Generalized Energy-Momentum Method. Several numerical examples illustrate the performance of the suggested contact formulation.

  • A primal-dual active set strategy for unilateral non-linear dynamic contact problems of thin-walled structures
    III European Conference on Computational Mechanics, 2006
    Co-Authors: Stefan Hartmann, Ekkehard Ramm, Stephan Brunssen, Barbara Wohlmuth

    Abstract:

    The efficient modeling of 3D contact problems is still a challenge in non-linear implicit structural analysis. Most of the existing contact algorithms use penalty methods to satisfy the contact constraints, which necessitates a user defined penalty parameter. As it is well known, the choice of this additional parameter is somehow arbitrary, problem dependent and influences the accuracy of the analysis. We use a primal-dual active set strategy [1], based on dual Lagrange multipliers [4] to handle the nonlinearity of the contact Conditions. This allows us to enforce the contact constraints in a weak, integral sense without any additional parameter. Due to the Biorthogonality Condition of the basis functions, the Lagrange multipliers can be locally eliminated. We perform a static condensation to get a reduced system for the displacements. The Lagrange multipliers, representing the contact pressure, can be easily recovered from the displacements in a variationally consistent way.

Barbara Wohlmuth – 3rd expert on this subject based on the ideXlab platform

  • Unilateral non‐linear dynamic contact of thin‐walled structures using a primal‐dual active set strategy
    International Journal for Numerical Methods in Engineering, 2020
    Co-Authors: Stefan Hartmann, Ekkehard Ramm, Stephan Brunssen, Barbara Wohlmuth

    Abstract:

    The efficient modelling of three-dimensional contact problems is still a challenge in non-linear implicit structural analysis. We use a primal-dual active set strategy (SIAM J. Optim. 2003; 13:865-888), based on dual Lagrange multipliers (SIAM J. Numer. Anal. 2000; 38:989-1012) to handle the non-linearity of the contact Conditions. This allows us to enforce the contact constraints in a weak, integral sense without any additional parameter. Due to the Biorthogonality Condition of the basis functions, the Lagrange multipliers can be locally eliminated. We perform a static condensation to achieve a reduced system for the displacements. The Lagrange multipliers, representing the contact pressure, can be easily recovered from the displacements in a variationally consistent way. For the application to thin-walled structures we adapt a three-dimensional non-linear shell formulation including the thickness stretch of the shell to contact problems. A reparametrization of the geometric description of the shell body gives us a surface-oriented shell element, which allows the application of contact Conditions directly to nodes lying on the contact surface. Shell typical locking phenomena are treated with the enhanced-assumed-strain-method and the assumed-natural-strain-method. The discretization in time is done with the implicit Generalized-α method (J. Appl. Mech. 1993; 60:371-375) and the Generalized Energy-Momentum Method (Comp. Methods Appl. Mech. Eng. 1999; 178:343-366) to compare the development of energies within a frictionless contact description. In order to conserve the total energy within the discretized frictionless contact framework, we follow an approach from Laursen and Love (Int. J. Numer. Methods Eng. 2002; 53:245-274), who introduced a discrete contact velocity to update the velocity field in a post-processing step. Various examples show the good performance of the primal-dual active set strategy applied to the implicit dynamic analysis of thin-walled structures.

  • A primal-dual active set strategy for unilateral non-linear dynamic contact problems of thin-walled structures
    III European Conference on Computational Mechanics, 2006
    Co-Authors: Stefan Hartmann, Ekkehard Ramm, Stephan Brunssen, Barbara Wohlmuth

    Abstract:

    The efficient modeling of 3D contact problems is still a challenge in non-linear implicit structural analysis. Most of the existing contact algorithms use penalty methods to satisfy the contact constraints, which necessitates a user defined penalty parameter. As it is well known, the choice of this additional parameter is somehow arbitrary, problem dependent and influences the accuracy of the analysis. We use a primal-dual active set strategy [1], based on dual Lagrange multipliers [4] to handle the nonlinearity of the contact Conditions. This allows us to enforce the contact constraints in a weak, integral sense without any additional parameter. Due to the Biorthogonality Condition of the basis functions, the Lagrange multipliers can be locally eliminated. We perform a static condensation to get a reduced system for the displacements. The Lagrange multipliers, representing the contact pressure, can be easily recovered from the displacements in a variationally consistent way.