scholarly journals Immersed boundary-conformal isogeometric method for linear elliptic problems

2021 ◽  
Author(s):  
Xiaodong Wei ◽  
Benjamin Marussig ◽  
Pablo Antolin ◽  
Annalisa Buffa
Keyword(s):  
Geometric Model ◽  
Boundary Representation ◽  
Dirichlet Boundary Conditions ◽  
Benchmark Problems ◽  
Immersed Boundary ◽  
Exterior Region ◽  
Conformal Method ◽  
Mesh Resolution ◽  
Background Mesh ◽  
Improved Accuracy

AbstractWe present a novel isogeometric method, namely the Immersed Boundary-Conformal Method (IBCM), that features a layer of discretization conformal to the boundary while employing a simple background mesh for the remaining domain. In this manner, we leverage the geometric flexibility of the immersed boundary method with the advantages of a conformal discretization, such as intuitive control of mesh resolution around the boundary, higher accuracy per degree of freedom, automatic satisfaction of interface kinematic conditions, and the ability to strongly impose Dirichlet boundary conditions. In the proposed method, starting with a boundary representation of a geometric model, we extrude it to obtain a corresponding conformal layer. Next, a given background B-spline mesh is cut with the conformal layer, leading to two disconnected regions: an exterior region and an interior region. Depending on the problem of interest, one of the two regions is selected to be coupled with the conformal layer through Nitsche’s method. Such a construction involves Boolean operations such as difference and union, which therefore require proper stabilization to deal with arbitrarily cut elements. In this regard, we follow our precedent work called the minimal stabilization method (Antolin et al in SIAM J Sci Comput 43(1):A330–A354, 2021). In the end, we solve several 2D benchmark problems to demonstrate improved accuracy and expected convergence with IBCM. Two applications that involve complex geometries are also studied to show the potential of IBCM, including a spanner model and a fiber-reinforced composite model. Moreover, we demonstrate the effectiveness of IBCM in an application that exhibits boundary-layer phenomena.

scholarly journals A numerical study of fish adaption behaviors in complex environments with a deep reinforcement learning and immersed boundary–lattice Boltzmann method

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Yi Zhu ◽  
Fang-Bao Tian ◽  
John Young ◽  
James C. Liao ◽  
Joseph C. S. Lai
Keyword(s):  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Prey Capture ◽  
Numerical Study ◽  
Benchmark Problems ◽  
Immersed Boundary ◽  
Complex Environments ◽  
Fish Model ◽  
Point To Point ◽  
Boltzmann Method

AbstractFish adaption behaviors in complex environments are of great importance in improving the performance of underwater vehicles. This work presents a numerical study of the adaption behaviors of self-propelled fish in complex environments by developing a numerical framework of deep learning and immersed boundary–lattice Boltzmann method (IB–LBM). In this framework, the fish swimming in a viscous incompressible flow is simulated with an IB–LBM which is validated by conducting two benchmark problems including a uniform flow over a stationary cylinder and a self-propelled anguilliform swimming in a quiescent flow. Furthermore, a deep recurrent Q-network (DRQN) is incorporated with the IB–LBM to train the fish model to adapt its motion to optimally achieve a specific task, such as prey capture, rheotaxis and Kármán gaiting. Compared to existing learning models for fish, this work incorporates the fish position, velocity and acceleration into the state space in the DRQN; and it considers the amplitude and frequency action spaces as well as the historical effects. This framework makes use of the high computational efficiency of the IB–LBM which is of crucial importance for the effective coupling with learning algorithms. Applications of the proposed numerical framework in point-to-point swimming in quiescent flow and position holding both in a uniform stream and a Kármán vortex street demonstrate the strategies used to adapt to different situations.

On the Volume Conservation of the Immersed Boundary Method

2012 ◽  
Vol 12 (2) ◽  
pp. 401-432 ◽  
Author(s):  
Boyce E. Griffith
Keyword(s):  
Projection Method ◽  
Projection Methods ◽  
Staggered Grid ◽  
Benchmark Problems ◽  
Immersed Boundary ◽  
Lagrangian Description ◽  
Volume Conservation ◽  
Modest Improvement ◽  
Order Of Magnitude ◽  
Ib Method

AbstractThe immersed boundary (IB) method is an approach to problems of fluid-structure interaction in which an elastic structure is immersed in a viscous incompressible fluid. The IB formulation of such problems uses a Lagrangian description of the structure and an Eulerian description of the fluid. It is well known that some versions of the IB method can suffer from poor volume conservation. Methods have been introduced to improve the volume-conservation properties of the IB method, but they either have been fairly specialized, or have used complex, nonstandard Eulerian finite-difference discretizations. In this paper, we use quasi-static and dynamic benchmark problems to investigate the effect of the choice of Eulerian discretization on the volume-conservation properties of a formally second-order accurate IB method. We consider both collocated and staggered-grid discretization methods. For the tests considered herein, the staggered-grid IB scheme generally yields at least a modest improvement in volume conservation when compared to cell-centered methods, and in many cases considered in this work, the spurious volume changes exhibited by the staggered-grid IB method are more than an order of magnitude smaller than those of the collocated schemes. We also compare the performance of cell-centered schemes that use either exact or approximate projection methods. We find that the volume-conservation properties of approximate projection IB methods depend strongly on the formulation of the projection method. When used with the IB method, we find that pressure-free approximate projection methods can yield extremely poor volume conservation, whereas pressure-increment approximate projection methods yield volume conservation that is nearly identical to that of a cell-centered exact projection method.

Construction of Analysis-Suitable Vascular Models Using Axis-Aligned Polycubes

2019 ◽  
Vol 141 (6) ◽  
Author(s):  
Adam R. Updegrove ◽  
Shawn C. Shadden ◽  
Nathan M. Wilson
Keyword(s):  
Medical Image ◽  
Geometric Model ◽  
Image Data ◽  
Boundary Representation ◽  
Patient Specific ◽  
Surface Model ◽  
Suitable Model ◽  
Computational Domain ◽  
Discrete Surface ◽  
Medical Image Data

Image-based modeling is an active and growing area of biomedical research that utilizes medical imaging to create patient-specific simulations of physiological function. Under this paradigm, anatomical structures are segmented from a volumetric image, creating a geometric model that serves as a computational domain for physics-based modeling. A common application is the segmentation of cardiovascular structures to numerically model blood flow or tissue mechanics. The segmentation of medical image data typically results in a discrete boundary representation (surface mesh) of the segmented structure. However, it is often desirable to have an analytic representation of the model, which facilitates systematic manipulation. For example, the model then becomes easier to union with a medical device, or the geometry can be virtually altered to test or optimize a surgery. Furthermore, to employ increasingly popular isogeometric analysis (IGA) methods, the parameterization must be analysis suitable. Converting a discrete surface model to an analysis-suitable model remains a challenge, especially for complex branched structures commonly encountered in cardiovascular modeling. To address this challenge, we present a framework to convert discrete surface models of vascular geometries derived from medical image data into analysis-suitable nonuniform rational B-splines (NURBS) representation. This is achieved by decomposing the vascular geometry into a polycube structure that can be used to form a globally valid parameterization. We provide several practical examples and demonstrate the accuracy of the methods by quantifying the fidelity of the parameterization with respect to the input geometry.

Boundary Evaluation for a Cellular Model

2003 ◽  
Author(s):  
Rafael Bidarra ◽  
Willem J. Neels ◽  
Willem F. Bronsvoort
Keyword(s):  
Geometric Model ◽  
Feature Modeling ◽  
Boundary Representation ◽  
Cellular Model ◽  
Performance Measurements ◽  
Shape Information ◽  
Cellular Models ◽  
Boundary Evaluation ◽  
Modeling Systems ◽  
Major Operation

Feature modeling systems usually employ a boundary representation (b-rep) to store the shape information on a product. It has, however, been shown that a b-rep has a number of shortcomings, and that a cellular model can be a valuable alternative. A cellular model stores additional shape information on a feature, including the faces that are not on the boundary of the product. Such information can be profitably used for several purposes. A major operation in each feature modeling system is boundary evaluation, which computes the geometric model of a product, i.e. either the b-rep or the cellular model, from the features that have been specified by the user. Because it has to be executed each time a feature has been added, removed or modified, its efficiency is very important. In this paper, boundary evaluation for a cellular model is described. Subsequently, its efficiency is compared to the efficiency of boundary evaluation for a b-rep, on the basis of performance measurements and considerations for both. It turns out that boundary evaluation for a cellular model is in fact more efficient than for a b-rep, which makes cellular models even more attractive as an alternative for b-reps.

Implicit Boundary Approach for Reissner-Mindlin Plates

2013 ◽  
Author(s):  
Hailong Chen ◽  
Ashok V. Kumar
Keyword(s):  
Boundary Conditions ◽  
Plate Theory ◽  
Weak Form ◽  
Mindlin Plate ◽  
Benchmark Problems ◽  
Element Analysis ◽  
Boundary Method ◽  
Solid Models ◽  
Plate Elements ◽  
Background Mesh

Implicit boundary method enables the use of background mesh to perform finite element analysis while using solid models to represent the geometry. This approach has been used in the past to model 2D and 3D structures. Thin plate or shell-like structures are more challenging to model. In this paper, the implicit boundary method is shown to be effective for plate elements modeled using Reissner-Mindlin plate theory. This plate element uses a mixed formulation and discrete collocation of shear stress field to avoid shear locking. The trial and test functions are constructed by utilizing approximate step functions such that the boundary conditions are guaranteed to be satisfied. The incompatibility of discrete collocation with implicit boundary approach is overcome by using irreducible weak form for computing the stiffness associated with essential boundary conditions. A family of Reissner-Mindlin plate elements is presented and evaluated in this paper using several benchmark problems to test their validity and robustness.

scholarly journals Application of Energetic BEM to 2D Elastodynamic Soft Scattering Problems

2019 ◽  
Vol 10 (1) ◽  
pp. 182-198
Author(s):  
A. Aimi ◽  
L. Desiderio ◽  
M. Diligenti ◽  
C. Guardasoni
Keyword(s):  
Wave Propagation ◽  
Dirichlet Boundary ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Dirichlet Boundary Conditions ◽  
Benchmark Problems ◽  
Weak Formulation ◽  
Scattering Problems ◽  
Propagation Analysis ◽  
First Time

Abstract Starting from a recently developed energetic space-time weak formulation of the Boundary Integral Equations related to scalar wave propagation problems, in this paper we focus for the first time on the 2D elastodynamic extension of the above wave propagation analysis. In particular, we consider elastodynamic scattering problems by open arcs, with vanishing initial and Dirichlet boundary conditions and we assess the efficiency and accuracy of the proposed method, on the basis of numerical results obtained for benchmark problems having available analytical solution.

scholarly journals A Comparison of Angular Discretization Techniques for the Radiative Transport Equation

2015 ◽  
Author(s):  
John Tencer
Keyword(s):  
Spherical Harmonics ◽  
Radiation Transport ◽  
Radiative Flux ◽  
Benchmark Problems ◽  
Discrete Ordinates ◽  
Flux Divergence ◽  
Discontinuous Boundary ◽  
All Solutions ◽  
Mesh Resolution ◽  
Ray Effects

Two of the most popular deterministic radiation transport methods for treating the angular dependence of the radiative intensity for heat transfer: the discrete ordinates and simplified spherical harmonics approximations are compared. A problem with discontinuous boundary conditions is included to evaluate ray effects for discrete ordinates solutions. Mesh resolution studies are included to ensure adequate convergence and evaluate the effects of the contribution of false scattering. All solutions are generated using finite element spatial discretization. Where applicable, any stabilization used is included in the description of the approximation method or the statement of the governing equations. A previous paper by the author presented results for a set of 2D benchmark problems for the discrete ordinates method using the PN-TN quadrature of orders 4, 6, and 8 as well as the P1, M1, and SP3 approximations. This paper expands that work to include the Lathrop-Carlson level symmetric quadrature of order up to 20 as well as the Lebedev quadrature of order up to 76 and simplified spherical harmonics of odd orders from 1 to 15. Two 3D benchmark problems are considered here. The first is a canonical problem of a cube with a single hot wall. This case is used primarily to demonstrate the potentially unintuitive interaction between mesh resolution, quadrature order, and solution error. The second case is meant to be representative of a pool fire. The temperature and absorption coefficient distributions are defined analytically. In both cases, the relative error in the radiative flux or the radiative flux divergence within a volume is considered as the quantity of interest as these are the terms that enter into the energy equation. The spectral dependence of the optical properties and the intensity is neglected.

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