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On time integration in the XFEM ; Zilian, Andreas in International Journal for Numerical Methods in Engineering (2009), 79(1), 69-93 The extended finite element method (XFEM) is often used in applications that involve moving interfaces. Examples are the propagation of cracks or the movement of interfaces in two-phase problems. This ... [more ▼] The extended finite element method (XFEM) is often used in applications that involve moving interfaces. Examples are the propagation of cracks or the movement of interfaces in two-phase problems. This work focuses on time integration in the XFEM. The performance of the discontinuous Galerkin method in time (space-time finite elements (FEs)) and time-stepping schemes are analyzed by convergence studies for different model problems. It is shown that space-time FE achieve optimal convergence rates. Special care is required for time stepping in the XFEM due to the time dependence of the enrichment functions. In each time step, the enrichment functions have to be evaluated at different time levels. This has important consequences in the quadrature used for the integration of the weak form. A time-stepping scheme that leads to optimal or only slightly sub-optimal convergence rates is systematically constructed in this work. © 2009 John Wiley & Sons, Ltd. [less ▲] Detailed reference viewed: 75 (3 UL)Space-Time Meshfree Collocation Method for PDEs ; Zilian, Andreas in PAMM (2009), 9(1), 651--652 An innovative Space-Time Meshfree Collocation Method (STMCM) for solving systems of nonlinear ordinary and partial differential equations by a consistent discretization in both space and time is proposed ... [more ▼] An innovative Space-Time Meshfree Collocation Method (STMCM) for solving systems of nonlinear ordinary and partial differential equations by a consistent discretization in both space and time is proposed as an alternative to established mesh-based methods. The STMCM belongs to the class of truly meshfree methods, i.e. the methods which do not have any underlying mesh, but work on a set of nodes only, without an a priori node-to-node connectivity. A regularization technique to overcome the singularity-by-construction and to compute all necessary derivatives of the kernel functions is presented. The method combines the simplicity and straightforwardness of the strong-form computational techniques with the advantages of meshfree methods over the classical ones, especially for coupled engineering problems involving moving interfaces. The key features of the proposed approach are: (i) no need to generate a mesh, (ii) simplified imposition of boundary conditions, (iii) no need to evaluate integral forms of governing equations, (iv) applicability to complex irregularly-shaped domains. The proposed STMCM is applied to linear and nonlinear ordinary and partial differential equations of different types and its accuracy and convergence properties are studied. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [less ▲] Detailed reference viewed: 128 (0 UL)A localized mixed-hybrid method for imposing interfacial constraints in the extended finite element method (XFEM) Zilian, Andreas ; in International Journal for Numerical Methods in Engineering (2009), 79(6), 733-752 The paper proposes an approach for the imposition of constraints along moving or fixed immersed interfaces in the context of the extended finite element method. An enriched approximation space enables ... [more ▼] The paper proposes an approach for the imposition of constraints along moving or fixed immersed interfaces in the context of the extended finite element method. An enriched approximation space enables consistent representation of strong and weak discontinuities in the solution fields along arbitrarily-shaped material interfaces using an unfitted background mesh. The use of Lagrange multipliers or penalty methods is circumvented by a localized mixed hybrid formulation of the model equations. In a defined region in the vicinity of the interface, the original problem is re-stated in its auxiliary formulation. The availability of the auxiliary variable enables the consideration of a variety of interface constraints in the weak form. The contribution discusses the weak imposition of Dirichlet- and Neumann-type interface conditions as well as continuity requirements not fulfilled a priori by the enriched approximation. The properties of the proposed approach applied to two-dimensional linear scalar- and vector-valued elliptic problems are investigated by studying the convergence behavior. © 2009 John Wiley & Sons,Ltd. [less ▲] Detailed reference viewed: 86 (0 UL)Space-time meshfree collocation method: Methodology and application to initial-boundary value problems ; Zilian, Andreas in International Journal for Numerical Methods in Engineering (2009), 80(3), 355-380 A novel space-time meshfree collocation method (STMCM) for solving systems of non-linear ordinary and partial differential equations by a consistent discretization in both space and time is proposed as an ... [more ▼] A novel space-time meshfree collocation method (STMCM) for solving systems of non-linear ordinary and partial differential equations by a consistent discretization in both space and time is proposed as an alternative to established mesh-based methods. The STMCM belongs to the class of truly meshfree methods, i.e. the methods that do not have any underlying mesh, but work on a set of nodes only without any a priori node-to-node connectivity. Instead, the neighbouring information is established on-the-fly. The STMCM is constructed using the Interpolating Moving Least-squares technique, which allows a simplified implementation of boundary conditions due to fulfillment of the Kronecker delta property by the kernel functions, which is not the case for the major part of other meshfree methods. The method is validated by several examples ranging from interpolation problems to the solution of PDEs, whereas the STMCM solutions are compared with either analytical or reference ones. © 2009 John Wiley & Sons, Ltd. [less ▲] Detailed reference viewed: 132 (4 UL)Projection-based reduction of fluid-structure interaction systems using monolithic space-time modes Zilian, Andreas ; ; in Computer Methods in Applied Mechanics & Engineering (2009), 198(47-48), 3795-3805 The focus of this work is the development of reduced models for engineering applications in complex bidirectional fluid-structure interaction. In the simultaneous solution procedure, velocity variables ... [more ▼] The focus of this work is the development of reduced models for engineering applications in complex bidirectional fluid-structure interaction. In the simultaneous solution procedure, velocity variables are used for both fluid and solid, and the whole set of model equations is discretized by a stabilized time-discontinuous space-time finite element method. Flexible structures are modeled using a three-dimensional continuum approach in a total Lagrangian setting considering large displacements and rotations. In the flow domain the incompressible Navier-Stokes equations describe the Newtonian fluid. A continuous finite element mesh is applied to the entire spatial domain, and the discretized model equations are assembled in a single set of algebraic equations, considering the two-field problem as a whole. The continuous fluid-structure mesh with identical orders of approximation for both solid and fluid in space and time automatically yields conservation of mass, momentum and energy at the fluid-structure interface. A mesh-moving scheme is used to adapt the nodal coordinates of the fluid space-time finite element mesh to the structural deformation. The computational approach for strongly coupled fluid-structure interaction is used to create suitable reduced models of generic nonlinear problems. Reduction is performed with monolithic projection-based space-time modes, ensuring strong coupling of fluid and structure in the reduced model. The contribution discusses results using proper orthogonal decomposition (POD) for determination of monolithic space-time modes in the reduction of fluid-structure systems. © 2009 Elsevier B.V. All rights reserved. [less ▲] Detailed reference viewed: 104 (3 UL)Enriched space-time finite elements for fluid-structure interaction ; Zilian, Andreas in European Journal of Computational Mechanics (2008), 17(5-7), 725-736 Detailed reference viewed: 73 (2 UL)Numerical model for tsunami generation by subaerial landslides ; Zilian, Andreas ; in PAMM (2008), 8(1), 10519--10520 A discretization method based on stabilized space–time finite elements is presented for the numerical analysis of three–fluid flows of immiscible and incompressible fluids. Signed distance functions are ... [more ▼] A discretization method based on stabilized space–time finite elements is presented for the numerical analysis of three–fluid flows of immiscible and incompressible fluids. Signed distance functions are used to assign the material properties to each spatial point in the domain. The motion and the change in topology of fluid–fluid interfaces are implicitly described by the level–set method. Strong and weak discontinuities in the fields of the physical state variables are captured by locally enriched approximations based on the partition–of–unity concept. An interior penalty method enforces interfacial conservation of mass and momentum. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [less ▲] Detailed reference viewed: 141 (1 UL)Weiterentwicklung eines mechanischen Modells zur Beschreibung Regen-Wind induzierter Schwingungen ; Zilian, Andreas ; in PAMM (2008), 8(1), 10883--10884 Detailed reference viewed: 108 (0 UL)The enriched space-time finite element method (EST) for simultaneous solution of fluid-structure interaction Zilian, Andreas ; in International Journal for Numerical Methods in Engineering (2008), 75(3), 305-334 The paper introduces a weighted residual-based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures. The presented method enables one to treat strongly ... [more ▼] The paper introduces a weighted residual-based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures. The presented method enables one to treat strongly coupled systems involving large structural motion and deformation of multiple-flow-immersed solid objects. The fluid flow is described by the incompressible Navier-Stokes equations. The current configuration of the thin structure of linear elastic material with non-linear kinematics is mapped to the flow using the zero iso-contour of an updated level set function. The formulation of fluid, structure and coupling conditions uniformly uses velocities as unknowns. The integration of the weak form is performed on a space-time finite element discretization of the domain. Interfacial constraints of the multi-field problem are ensured by distributed Lagrange multipliers. The proposed formulation and discretization techniques lead to a monolithic algebraic system, well suited for strongly coupled fluid-structure systems. Embedding a thin structure into a flow results in non-smooth fields for the fluid. Based on the concept of the extended finite element method, the space-time approximations of fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions. This leads to the present enriched space-time (EST) method. Numerical examples of fluid-structure interaction show the eligibility of the developed numerical approach in order to describe the behavior of such coupled systems. The test cases demonstrate the application of the proposed technique to problems where mesh moving strategies often fail. Copyright © 2007 John Wiley & Sons, Ltd. [less ▲] Detailed reference viewed: 125 (6 UL)Simulation der aktiven Schwingungskontrolle von Fluid-Struktur Wechselwirkung durch piezoelektrische Materialien ; ; Zilian, Andreas in PAMM (2008), 8(1), 10529--10530 Ein numerisches Modell zur aktiven Kontrolle des dynamischen Verhaltens von Tragwerken unter transienter Windeinwirkung wird vorgestellt. Auf der Oberschicht passiver Tragstrukturen werden ... [more ▼] Ein numerisches Modell zur aktiven Kontrolle des dynamischen Verhaltens von Tragwerken unter transienter Windeinwirkung wird vorgestellt. Auf der Oberschicht passiver Tragstrukturen werden piezoelektrische Sensoren und Aktoren appliziert. Die resultierenden adaptierbaren Strukturen sind mit einem Regler gekoppelt. Die Modellierung der elektroelastischen Struktur erfolgt mit der geometrisch nichtlinearen Elastizitätstheorie und der Maxwell–Faraday–Theorie der Elektrostatik. Die Windumströmung der Struktur ist als inkompressibles, Newton'sches Fluid mit den Navier–Stokes–Gleichungen beschrieben. Die Diskretisierung der gekoppelten Modellgleichungen beider Kontinua erfolgt mit finiten Raum–Zeit–Elementen. Zeitliche Änderungen der Zustandsgrößen werden mit dem diskontinuierlichen Galerkin–Verfahren approximiert. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [less ▲] Detailed reference viewed: 100 (1 UL) |
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