Isoparametric B-Spline Elements for Immersed Boundary Explicit Dynamic Simulation

2020 ◽  
Vol 20 (4) ◽  
Author(s):  
Shashank Menon ◽  
Ashok V. Kumar
Keyword(s):  
Boundary Conditions ◽  
Dynamic Simulation ◽  
Small Time ◽  
Immersed Boundary ◽  
Element Analysis ◽  
B Spline ◽  
B Splines ◽  
Background Mesh ◽  
Definition Of ◽  
Explicit Dynamic

Abstract Explicit dynamic analysis has proven to be advantageous when simulating shock and impact loading, and very small time-scale events. In this article, the feasibility of using a background mesh of B-spline elements for immersed boundary explicit dynamic simulation is studied. In this approach, the geometry is immersed in a background mesh consisting of uniform regular shaped elements to avoid mesh generation difficulties. The boundary conditions are applied using the step boundary method, which uses the equations of the boundaries to construct trial functions that satisfy the essential boundary conditions. An isoparametric formulation is presented for quadratic and cubic B-spline elements and their shape functions are derived from the classical recursive definition of B-splines. The effectiveness of mass diagonalization for B-spline elements is also explored. This approach is validated using several examples by comparing with modal superposition solutions as well as past work using traditional finite element analysis (FEA) and analytical solutions when available.

scholarly journals Shape Sensing of a Complex Aeronautical Structure with Inverse Finite Element Method

Sensors ◽  
2021 ◽  
Vol 21 (4) ◽  
pp. 1388
Author(s):  
Daniele Oboe ◽  
Luca Colombo ◽  
Claudio Sbarufatti ◽  
Marco Giglio
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Conditions ◽  
Strain Field ◽  
Element Analysis ◽  
Inverse Finite Element Method ◽  
Shape Sensing ◽  
Definition Of ◽  
High Level ◽  
Element Method

The inverse Finite Element Method (iFEM) is receiving more attention for shape sensing due to its independence from the material properties and the external load. However, a proper definition of the model geometry with its boundary conditions is required, together with the acquisition of the structure’s strain field with optimized sensor networks. The iFEM model definition is not trivial in the case of complex structures, in particular, if sensors are not applied on the whole structure allowing just a partial definition of the input strain field. To overcome this issue, this research proposes a simplified iFEM model in which the geometrical complexity is reduced and boundary conditions are tuned with the superimposition of the effects to behave as the real structure. The procedure is assessed for a complex aeronautical structure, where the reference displacement field is first computed in a numerical framework with input strains coming from a direct finite element analysis, confirming the effectiveness of the iFEM based on a simplified geometry. Finally, the model is fed with experimentally acquired strain measurements and the performance of the method is assessed in presence of a high level of uncertainty.

On the Literature of B-Spline Interpolation Functions

2009 ◽  
pp. 214-222
Author(s):  
Carlo Ciulla
Keyword(s):  
Scientific Community ◽  
Polynomial Interpolation ◽  
Spline Interpolation ◽  
Spline Functions ◽  
Great Effort ◽  
Formal Definition ◽  
B Spline ◽  
B Splines ◽  
Scientific Disciplines ◽  
Definition Of

This chapter reviews the extensive and comprehensive literature on B-Splines. In the forthcoming text emphasis is given to hierarchy and formal definition of polynomial interpolation with specific focus to the subclass of functions that are called B-Splines. Also, the literature is reviewed with emphasis on methodologies and applications of B-Splines within a wide array of scientific disciplines. The review is conducted with the intent to inform the reader and also to acknowledge the merit of the scientific community for the great effort devoted to B-Splines. The chapter concludes emphasizing on the proposition that the unifying theory presented throughout this book has for what concerns two specific cases of B-Spline functions: univariate quadratic and cubic models.

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.

Definition of Boundary Conditions and Dynamic Analysis of Rocket Sled and Turntable

2011 ◽  
Vol 52-54 ◽  
pp. 261-266 ◽  
Author(s):  
Jian Hua Zhang ◽  
Shou Shan Jiang
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Boundary Conditions ◽  
Multibody System ◽  
Simulation Method ◽  
Reliable Data ◽  
Dynamics Analysis ◽  
Element Analysis ◽  
Analysis Theory ◽  
Definition Of

The Dynamics Analysis & Simulation of the Rocket Sled were done based on the Multibody System Dynamics and Finite Element Analysis Theory. The most difficult work in the analysis is how to establish the boundary conditions of the rocket sled. This paper makes this kind of attempt. Then the relevant postprocessing figures and data were obtained，thereby providing the designer and manufacturer with detailed and reliable data. The conclusion is the simulation method is more effective than those before and the boundary conditions are acceptable.

scholarly journals Analysis and Application of Quadratic B-spline Interpolation for Boundary Value Problems

2020 ◽  
pp. 11-21
Author(s):  
Md. Asaduzzaman ◽  
Liton Chandra Roy ◽  
Md. Musa Miah
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Spline Interpolation ◽  
Boundary Value ◽  
Data Interpolation ◽  
B Spline ◽  
B Splines ◽  
Numerical Example

B-splines interpolations are very popular tools for interpolating the differential equations under boundary conditions which were pioneered by Maria et.al.[16] allowing us to approximate the ordinary differential equations (ODE). The purpose of this manuscript is to analyze and test the applicability of quadratic B-spline in ODE with data interpolation, and the solving of boundary value problems. A numerical example has been given and the error in comparison with the exact value has been shown in tabulated form, and also graphical representations are shown. Maple soft and MATLAB 7.0 are used here to calculate the numerical results and to represent the comparative graphs.

scholarly journals Uniform B-Splines for the VTK Imaging Pipeline

2011 ◽  
Author(s):  
David Gobbi ◽  
Yves P. Starreveld
Keyword(s):  
Image Processing ◽  
Boundary Conditions ◽  
Coordinate Transformation ◽  
Image Interpolation ◽  
Piecewise Polynomial ◽  
Image Deformation ◽  
B Spline ◽  
Different Boundary Conditions ◽  
B Splines

Uniform B-splines are used widely in image processing because they provide maximal smoothness compared to any other piecewise polynomial of the same degree and support. This paper describes VTK classes for performing two functions: image interpolation via B-splines, and non-rigid coordinate transformation via B-splines. Special attention is paid to different boundary conditions for the ends of the spline: image interpolation supports clamped, mirrored, and repeated boundary conditions while B- spline grid transformations support clamped and zero-past-boundary conditions. The use of these classes for image deformation is demonstrated.

A new algorithm based on modified trigonometric cubic B-splines functions for nonlinear Burgers’-type equations

2017 ◽  
Vol 27 (8) ◽  
pp. 1638-1661 ◽  
Author(s):  
Ram Jiwari ◽  
Ali Saleh Alshomrani
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Spline Functions ◽  
Difference Operators ◽  
Content Type ◽  
B Spline ◽  
B Splines ◽  
Burgers Type Equations

Purpose The main aim of the paper is to develop a new B-splines collocation algorithm based on modified cubic trigonometric B-spline functions to find approximate solutions of nonlinear parabolic Burgers’-type equations with Dirichlet boundary conditions. Design/methodology/approach A modification is made in cubic trigonometric B-spline functions to handle the Dirichlet boundary conditions and an algorithm is developed with the help of modified cubic trigonometric B-spline functions. The proposed algorithm reduced the Burgers’ equations into a system of first-order nonlinear ordinary differential equations in time variable. Then, strong stability preserving Runge-Kutta 3rd order (SSP-RK3) scheme is used to solve the obtained system. Findings A different technique based on modified cubic trigonometric B-spline functions is proposed which is quite different from to the schemes developed in Abbas et al. (2014) and Nazir et al. (2016), and the developed algorithms are free from linearization process and finite difference operators. Originality/value To the best knowledge of the authors, this technique is novel for solving nonlinear partial differential equations, and the new proposed technique gives better results than the results discussed in Ozis et al. (2003), Kutluay et al. (1999), Khater et al. (2008), Korkmaz and Dag (2011), Kutluay et al. (2004), Rashidi et al. (2009), Mittal and Jain (2012), Mittal and Jiwari (2012), Mittal and Tripathi (2014), Xie et al. (2008) and Kadalbajoo et al. (2005).

Total Lagrangian Formulation for Large Deformation Modeling Using Uniform Background Mesh

2016 ◽  
Author(s):  
Nikhil Bhosale ◽  
Ashok V. Kumar
Keyword(s):  
Boundary Conditions ◽  
Large Deformation ◽  
Nonlinear Problems ◽  
Lagrangian Formulation ◽  
Element Analysis ◽  
Static And Dynamic Analysis ◽  
Main Challenge ◽  
Background Mesh ◽  
Uniform Background ◽  
Total Lagrangian Formulation

Mesh generation difficulties can be avoided when a background mesh rather than a mesh that conforms to the geometry is used for the analysis. The geometry is represented by equations and is independent of the mesh and is immersed in the background mesh. The solution to boundary value problems is approximated or piece-wise interpolated using the background mesh. The main challenge is in applying the boundary conditions because the boundaries may not have any nodes on them. Implicit boundary method has been used for linear static and dynamic analysis and has shown to be an effective approach for imposing boundary conditions but has never been applied to nonlinear problems. In this paper, this approach is extended to large deformation nonlinear analysis using the Total Lagrangian formulation. The equations are solved using the widely used modified Newton-Raphson method with loads applied over many load steps. Several test examples are studied and compared with traditional finite element analysis software for verification.

Survey of Immersed Boundary Approaches for Finite Element Analysis

2020 ◽  
Vol 20 (4) ◽  
Author(s):  
Ashok V. Kumar
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Mesh Generation ◽  
Immersed Boundary ◽  
Uniform Mesh ◽  
Element Analysis ◽  
Hexahedral Elements ◽  
Boundary Method ◽  
Background Mesh ◽  
Plates And Shells

Abstract Mesh generation for traditional finite element analysis has proven to be very difficult to fully automate especially using hexahedral elements for complex 3D geometry. Several modifications to the finite element method (FEM), such as the meshless methods, have been proposed for avoiding mesh generation. An alternative approach has recently gained popularity where the geometry, created as a solid model in cad software, is embedded or immersed in a nonconforming background mesh for analysis. In this approach, referred to here as the immersed boundary approach, a background mesh that is independent of the geometry is used for piecewise interpolation or approximation of the solution. Therefore, a uniform mesh with regular-shaped or undistorted elements can be used, and such a mesh is easy to generate automatically. When the geometry is immersed in the background mesh, the boundary elements are often only partly inside the geometry and the nodes of the mesh may not be on the boundaries. Many new methods have been developed to integrate over partial elements and to apply boundary and interface conditions when the boundaries of the geometries do not conform to the background mesh. These methods are reviewed in this article with particular emphasis on the implicit boundary method and step boundary method for applying boundary conditions. In addition, B-spline elements and several applications of the immersed boundary approach are surveyed including composite microstructures and structural elements for plates and shells.

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