Documents published in Scipedia

• Stabilization of mixed displacement-pressure finite elements at finite strains using polyhedral formulations and Voronoi meshing

  B. Sauren, E. Oheim, S. Klinkel

  WCCM2024.

  
  

  Abstract

  The hexahedral mixed displacement-pressure finite element of the lowest order (H1/P0) has shown to be simple and effective during both linear and nonlinear analysis of incompressible solids. While the discrete displacement field is generally considered to be sufficiently accurate, the discrete pressure field can sometimes be heavily polluted by spurious pressure modes. This results from the fact that the element does not fulfill the inf-sup condition. While postprocessing techniques, such as pressure filtering or smoothing, exist to remove the spurious pressure modes from the solution, this contribution aims on the exclusion of spurious pressure modes from the solution a priori due to the element geometry. By employing polyhedral finite element formulations on Voronoi tessellations in three dimensions, we show that the discrete kernel of the linearized mixed bilinear form only consists of the hydrostatic pressure mode. A spurious pressure mode is automatically suppressed due to the vertex-to-volume ratio in the finite element mesh. These considerations hold for any arbitrary physically admissible displacement state that can occur within a Newton-Raphson framework. A nonlinear numerical example shows that spurious pressure modes are indeed suppressed if the type of tessellation is changed from hexahedral to Voronoi.

  Abstract
  The hexahedral mixed displacement-pressure finite element of the lowest order (H1/P0) has shown to be simple and effective during both linear and nonlinear analysis of incompressible [...]

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• A coupled SPH-FDM method for simulations of unsteady flows

  C. Huang, H. Jiang, F. He, A. Shakibaeinia

  WCCM2024.

  
  

  Abstract

  The fluid-flexible-structure interaction (FFSI) is characterized by the large deformation, the thin structure, and the complex of the flow field. Accurately simulating FFSI poses three challenges, which are the reproduction of thin structure, the capture of moving interface, and the numerical stability of multi-physics field coupling, respectively. In this study, the FFSI is simulated by the smoothed particle hydrodynamics (SPH) because of its natural advantage in dealing with the moving interface. The shell model with single-layer particles[1] is introduced into SPH to simulate the thin flexible structure. The truncation error caused by the single-layer boundary is modified by the normal flux approach[2]. κ-ε turbulence model is introduced into SPH to enhance the numerical stability and capture complex flow details. In addition, other techniques or models that ensure the efficiency and stability of the calculation are used in this study, including PST (particle shifting technique), δ-SPH method, and GPU (graphics processing unit). The flows around the single filament are simulated to verify the accuracy and stability of the current FFSI algorithm based on the SPH method.

  Abstract
  The fluid-flexible-structure interaction (FFSI) is characterized by the large deformation, the thin structure, and the complex of the flow field. Accurately simulating FFSI [...]

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• A computationally efficient method for considering a large number of nonlinear multi-point constraints within the finite element method

  J. Wackerfuβ, J. Boungard

  WCCM2024.

  
  

  Abstract

  Nonlinear constraints are crucial in modeling various problems in computational mechanics. Among other things, they can be used for the subsequent consideration of rigid inclusions in a body originally modeled as deformable, without requiring a remeshing of the considered domain and thus contributing to a rapid modeling building. Unlike Lagrange multipliers and the penalty method, the master-slave elimination reduces the problem dimension but is limited to linear constraints. We introduce a new master-slave elimination method for arbitrary nonlinear multi-point constraints. It is compared to existing methods through analysis of the resulting equations and numerical examples. Results indicate that the method is as accurate, robust, and flexible as Lagrange multipliers, with improved efficiency due to reduced degrees of freedom, which is particularly advantageous when a large number of constraints have to be considered.

  Abstract
  Nonlinear constraints are crucial in modeling various problems in computational mechanics. Among other things, they can be used for the subsequent consideration of rigid inclusions [...]

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• Geometric formulation of three-temperature radiation hydrodynamics

  B. Tran, B. Southworth, J. Burby, M. Leok

  WCCM2024.

  
  

  Abstract

  Three-temperature (3T) radiation hydrodynamics models high energy-density plasma of nonlinearly coupled electron, ion, and radiation fields, finding applications in astrophysics and inertial confinement fusion. We present a geometric formulation of three-temperature radiation hydrodynamics. This is done utilizing an irreverisble portHamiltonian framework in the entropy representation. This geometric formulation separates the advection, interaction, and diffusion processes occuring into separate operators and establishes the energy-preserving interconnections between them. Structural properties such as mass, momentum and energy conservation as well as entropy production arise naturally from the geometric formulation. As an application, we briefly discuss a framework for the energy control of the 3T system within the port-Hamiltonian framework.

  Abstract
  Three-temperature (3T) radiation hydrodynamics models high energy-density plasma of nonlinearly coupled electron, ion, and radiation fields, finding applications in astrophysics [...]

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• The Picard-Newton iteration for the Boussinesq equations

  L. Rebholz, E. Hawkins

  WCCM2024.

  
  

  Abstract

  We consider the Picard-Newton and Anderson accelerated Picard-Newton solvers applied to the Boussinesq equations, nonlinear Helmholtz equations and Liouville equation, for the purpose of accelerating convergence and improving robustness with respect to problem parameters. In all cases, we show the proposed solvers improve efficiency over the commonly used solvers and are able to find solutions for a much larger set of problem parameters.

  Abstract
  We consider the Picard-Newton and Anderson accelerated Picard-Newton solvers applied to the Boussinesq equations, nonlinear Helmholtz equations and Liouville equation, for [...]

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• A new concept for embedding sub-structures via level-sets

  T. Fries, J. Neumeyer, M. Kaiser

  WCCM2024.

  
  

  Abstract

  A framework is presented to continuously embed sub-structures such as fibres and membranes into otherwise homogeneous, isotropic bulk materials. The bulk material is modeled with classical finite strain theory. The sub-structures are geometrically defined via all level sets of a scalar function over the bulk domain. A mechanical model that is simultaneously applicable to all level sets is given and coupled to the bulk material. This results in a new concept for anisotropic materials with possible applications in biological tissues, layered rocks, composites, and textiles. For the numerical analysis, the bulk domain is discretized possibly using higher-order finite elements which do not conform to the level sets implying the shapes of the embedded sub-structures. Numerical results confirm the success of the proposed embedded sub-structure models in different contexts

  Abstract
  A framework is presented to continuously embed sub-structures such as fibres and membranes into otherwise homogeneous, isotropic bulk materials. The bulk material is modeled [...]

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• On triangular self-stabilized virtual elements for Kirchhoff-Love shells

  T. Park Wu, P. de Mattos Pimenta, P. Wriggers

  WCCM2024.

  
  

  Abstract

  This work presents a self-stabilized triangular virtual element for linear Kirchhoff–Love shells. The domain decomposition by flat triangles directly approximates the shell geometry without resorting to a curvilinear coordinate system or an initial mapping approach. The problem is discretized by the lowest-order conventional virtual element method for the membrane, in which stabilization is needless, and by a stabilization-free virtual element procedure for the plate. Numerical examples of static problems show the potential of the formulation as a prelude for the evolution of self-stabilized Kirchhoff–Love shell virtual elements.

  Abstract
  This work presents a self-stabilized triangular virtual element for linear Kirchhoff–Love shells. The domain decomposition by flat triangles directly approximates the shell [...]

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• Projector assembly: bridging Poisson and elasticity formulations

  T. Moherdaui, A. Gay Neto, P. Wriggers

  WCCM2024.

  
  

  Abstract

  Virtual element methods define their shape functions implicitly (tailored to each element’s geometry), foregoing the typical reference element and transformation scheme usually employed by the finite element method. The formulation leverages the use of polynomial projections supplied by heuristic stabilizations when necessary. These projections are represented by projector matrices, which require the solution of a local system. Elasticity formulations usually employ an 𝐿 2 -projection from a displacement multifield onto a strain multifield, requiring the solution of a considerably larger system than a typical Poisson problem would require, with dense matrices and lots of zeroes. This work presents a way to obtain the projections for elasticity formulation by assembling from the 𝐿 2 -projection for each derivative of the one-field a Poisson formulation, resulting in smaller local systems being solved and more efficient storage. This approach is based on the linearity of both projections and derivatives, and is shown in the examples to preserve the convergence rate of the method.

  Abstract
  Virtual element methods define their shape functions implicitly (tailored to each element’s geometry), foregoing the typical reference element and transformation scheme [...]

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• Mixed partition of unity methods and stochastic Gillespie algorithms for Transport-Reaction equations

  M. Kirkilionis

  WCCM2024.

  
  

  Abstract

  We introduce a new type of model framework which is part stochastic and part deterministic. The starting point is a finite size particle system within a single reaction volume, with type exchanges modelled by a contact process. Inside the reaction volume, each particle can interact with every other particle with the same probability. This is the setting of a classical reaction system simulated with a Gillespie algorithm. Such systems can be used to describe other than chemistry type exchanges, like an infection process, and therefore are already very versatile. Their advantage is that they are able to be used where small size effects can play a role, like extinction events, which are impossible to model with differential equations, including stochastic differential equations. However finite size and single reaction volume settings for reaction systems are too restrictive in other ways. We might like to add internal or external states to the particles. These states are coordinates in a position space. An example of an internal position/state space is age (since entering the system), an example for an external position/state space is geographical location. The particles then can also change their positions in these state spaces, according to some probability distribution which evolution is modelled deterministically. The classical example for a transport process is a partial differential equation like the heat equation, or more general parabolic advection-diffusion equations. We assume that the distribution of the particles in position space is not influencing the evolution of the probability distribution driving in turn the evolution of the particles’ positions. The model framework with its finite-size particle population approach can very accurately model situations where finite-size effects take place, however provides in addition detailed descriptions of both internal and external particle state spaces where needed. The framework can therefore be used in addition to traditional established models, like transport PDEs or internally structured population models, when the computation of the statistics of finite-size effects is important.

  Abstract
  We introduce a new type of model framework which is part stochastic and part deterministic. The starting point is a finite size particle system within a single reaction volume, [...]

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• Immersed-boundary approach based on integrated RBFs and smooth extension for solving PDEs in complex domains

  N. Mai-Duy, C. Tran, D. Strunin, W. Karunasena

  WCCM2024.

  
  

  Abstract

  We propose an immersed-boundary approach, based on point collocation, five-point integrated radial basis function stencils, rectangular Cartesian grids and smooth extension of the solution, for solving the two-dimensional elliptic partial differential equation in a geometrically complex domain.

  Abstract
  We propose an immersed-boundary approach, based on point collocation, five-point integrated radial basis function stencils, rectangular Cartesian grids and smooth extension [...]

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