Isoparametric Mapping Research Articles

Isoparametric Functions and Solutions of Yamabe Type Equations on Manifolds with Boundary

Isoparametric Functions and Solutions of Yamabe Type Equations on Manifolds with Boundary

A linear smoothed quadratic finite element for buckling analysis of laminated composite plates

In this paper, a linear smoothing scheme over eight-node Reissner-Mindlin plate element under the framework of the CS-FEM is employed to buckling analysis of laminated composite plates based on the first-order shear deformation theory. The modified stain matrix is computed by the divergence theorem between the nodal shape functions and their derivatives using Taylor's expansion. Isoparametric mapping and the computation of interior derivatives of shape function is not required in the proposed method, furthermore, all the computation are based on the global Cartesian coordinates. Some numerical examples are given at the end to demonstrate that the present method has good performance to alleviate the shear-locking phenomenon and improve the quality of the solutions with distorted meshes.

Numerical analysis of thermal and mechanical characteristics with property maps in complex semiconductor package designs

As semiconductor performance improves through advanced package designs, it becomes important to consider both thermal and mechanical properties. Better understanding their relationship enhances system design and optimization. This study has introduced a numerical method that takes into account both thermal and mechanical fluxes to construct thermal and mechanical property maps for practical application in semiconductor engineering. Furthermore, this paper investigated the relationship between the two properties using the constructed property maps in commercially available semiconductor packages. Owing to the complexity of packaging design patterns, a coupled isoparametric mapping and machine learning (ML) method was introduced to investigate their effects. The model then determines and investigates the anisotropic equivalent thermal and mechanical properties of the commercialized semiconductor packages with complex pattern designs, according to the package materials, volume fractions of each material, and design patterns. This approach pursues more sophistication and practicality compared to simplified composite structure property evaluation models. Additionally, all processes in the algorithm are automated based on ML methods, making them practically applicable in the semiconductor industry. The study shows that there is a strong correlation between the thermal and mechanical characteristics in the complex package patterns. This relationship was able to form a fairly clear category based on the shape of the dielectric band because the dielectric band had significantly different effects on the thermal and mechanical fluxes. The study discussed the physical implications of the similarities and differences between the two behaviors and validated the results of the equivalent properties through finite element (FE) analysis. Finally, the study cross-validates the relationship by showing that information from one flux behavior can be used to predict the other based on the ML method. Based on the results, this paper can provide a better understanding of thermal and mechanical fluxes in semiconductor package patterns for package design and optimization.

Stratigraphic and Petrophysical assessment of ASL sandstone reservoir in northern and central October Field, Gulf of Suez

ABSTRACT This study analysed the depositional environment and diagenesis of the ASL Member reservoir in the October oil field located in the Gulfof Suez Basin, Egypt. Sedimentological and petrophysical techniques were used to interpret data from four wells: OCT-J7B, GS173-2, GS184-4A, and GS195-3, penetrating various rock types, including sandstone, carbonate, and fractured basement reservoirs. Log correlations of OCT-J7B and GS173-2 showed they penetrated pre-Miocene strata, while GS184-4A and GS195-3 reached the Rudeis section. Based on seismic-based structural maps, all wells except GS173-2 were drilled down-dip of the main Clysmic fault. The Hawara shale and the older shales in the Rudeis section were found to be the reservoir’s lateral seal, and the Lagia shale was found to be its vertical seal. Core analysis of OCT-J7B revealed the ASL as limestone interbedded with shale and sandstone deposited in a proximal submarine fan lobe, while GS184-4A and GS195-3 were located in amid-fan carbonate system. The comparisons of core and log data from the four wells yielded excellent matches, and the iso-parametric mapping delineated reservoir quality horizontally and hydrocarbon potentiality. The study recommends sitting future wells away from uplifts and suggests further reservoir characterization and evaluation of potential zones in the ASL Member.

Isoparametric functions and mean curvature in manifolds with Zermelo navigation

Isoparametric functions and mean curvature in manifolds with Zermelo navigation

Local element operations for curved simplex meshes

SummaryMesh optimization procedures are generally a combination of node smoothing and discrete operations which affect a small number of elements to improve the quality of the overall mesh. These procedures are useful as a post‐processing step in mesh generation procedures and in applications such as fluid simulations with severely deforming domains. In order to perform high‐order mesh optimization, these ingredients must also be extended to high‐order (curved) meshes. In this work, we present a method to perform local element operations on curved meshes. The mesh operations discussed in this work are edge/face swaps, edge collapses, and edge splitting (more generally refinement) for triangular and tetrahedral meshes. These local operations are performed by first identifying the patch of elements which contain the edge/face being acted on, performing the operation as a “straight‐sided one" by placing the high‐order nodes via an isoparametric mapping from the master element, and smoothing the high‐order nodes on the elements in the patch by minimizing a Jacobian‐based high‐order mesh distortion measure. Since the initial “straight‐sided guess” from the placement of the nodes via the isoparametric mapping frequently results in invalid elements, the distortion measure must be regularized which allows for mesh untangling for the optimization to succeed. We present several examples in 2D and 3D to demonstrate these local operations and how they can be combined with a high‐order node smoothing procedure to maintain mesh quality when faced with severe deformations.

Comparative Study of Skin–Stringer Connection Approaches for Stiffened Structures with Curvilinear Stiffeners

Additive manufacturing enables the deposition of material in the near-net shape at any desired location on the base plate or shell to improve the structural performance by tailoring the buckling and/or vibration mode shapes. In this study, a stiffened plate with arbitrarily shaped stiffeners is modeled using a nonconformal mesh-based finite element model. Six degrees of freedom are considered for each node of the plate and the stiffener finite element models to enable the couplings between the plate’s and stiffeners’ in-plane and out-of-plane motions. Displacement compatibility is enforced at the interface between the plate and the stiffeners where several skin–stringer connection approaches, including inverse isoparametric mapping algorithm (IIMA), radial basis function, and thin-plate spline function, are used. Various complex models available in the literature are employed for evaluating the skin–stringer connection approaches. Research studies show that the IIMA-based nonconformal mesh modeling can generate a sparse displacement approximation matrix, which makes it an efficient approach for nonconformal mesh-based modeling. For the stiffened plate with higher stiffeners, the present results agree well with those obtained from the Abaqus/CAE conventional conformal mesh and mesh tie constraint-based nonconformal mesh modeling results, but yield better results than the Abaqus/CAE wrapping mesh approach.

An efficient strategy to implement local porosity constraints in the multiscale design of solids with parameterized biomimetic microstructures

In previous works, the authors introduced a multiscale optimization method to maximize the stiffness of elastic solids with biomimetic cancellous microstructures described by a finite set of parameters. Although effective, the procedure is computationally expensive when solving large-scale problems using per-element non-linear constraints to impose local bounds on the solid volume fraction. This work improves the computational performance of the method by exploring two strategies to completely dispense with nonlinear local constraints: to bound the microparameters so the microsctructures are always within the solid fraction of trabecular bone, and to map the microparameters onto an auxiliary set of parameters that are linearly bounded. As a side effect, the design spaces are reduced. Such reductions are assessed in terms of the bulk and shear moduli and elastic symmetries, which are compared to those of natural bone. Performances of the two strategies are assessed by solving a series of benchmark problems and studying the stiffness of a hip prosthesis. The strategy based on the isoparametric mapping achieves the best results, performing up to 2000 times faster while marginally reducing the design space. Thus, the isoparametric mapping approach makes the multiscale design method a suitable tool for solving large-scale problems of practical interest.

A cell-based smoothed finite element method for multi-body contact analysis within the bi-potential formulation

This paper presents a cell-based smoothed finite element method (CS-FEM) for solving contact problems within the bi-potential formulation. The frictional contact and large deformation are modeled based on the CS-FEM. This method performs based on the generalized smoothed Galerkin weak-form which requires only value of shape functions. Because no isoparametric mapping is needed, the CS-FEM has an advantage when dealing with problems with large deformation which may cause mesh distortion. The contact forces and the relative velocity establish an implicit relationship within the bi-potential formulation in which no user-defined parameter is introduced. Numerical examples show the contact between elastic and hyperelastic bodies calculated by CS-FEM models with different smoothing domains. Compared with FEM, the CS-FEM produces robust and more accurate solutions.

A stochastic finite element scheme for solving partial differential equations defined on random domains

This paper proposes a novel stochastic finite element scheme to solve partial differential equations defined on random domains. A geometric mapping algorithm first transforms the random domain into a reference domain. By combining the mesh topology (i.e. the node numbering and the element numbering) of the reference domain and random nodal coordinates of the random domain, random meshes of the original problem are obtained by only one mesh of the reference domain. In this way, the original problem is still discretized and solved on the random domain instead of the reference domain. A random isoparametric mapping of random meshes is then proposed to generate the stochastic finite element equation of the original problem. We adopt a weak-intrusive method to solve the obtained stochastic finite element equation. In this method, the unknown stochastic solution is decoupled into a sum of the products of random variables and deterministic vectors. Deterministic vectors are computed by solving deterministic finite element equations, and corresponding random variables are solved by a proposed sampling method. The computational effort of the proposed method does not increase dramatically as the stochastic dimension increases and it can solve high-dimensional stochastic problems with low computational effort, thus the proposed method avoids the curse of dimensionality successfully. Four numerical examples are given to demonstrate the good performance of the proposed method.

The isoparametric functions on a class of Finsler spheres

In this paper, we give global expressions of geodesics and isoparametric functions on a standard Randers sphere by navigation. We obtain isoparametric families and focal submanifolds in (Sn,F,dμBH) by Cartan-Münzner polynomials. Further more, we construct some examples of closed and non-closed geodesics, isoparametric functions, isoparametric families and focal submanifolds.

A High-Precision Interpolation Method for Data Transfer in Fluid-Structure Interaction Analysis

In order to improve the precision of data transfer in fluid-structure interaction (FSI) analysis, this paper puts forward an improved inverse isoparametric mapping (IIM) method, which can improve the load transfer accuracy by means of increasing the quantity of Gauss integral points. As displacement transfer happens, the method can perform the function of revising the displacement between the point to be interpolated and projection point, by which the example emulation is also carried out. Some practical solutions are presented, aiming to grapple with pertinent problems such as local interpolation burr and global interpolation distortion which are originated from oversimplified structure dynamic models and inconsistent structural deformation. The calculation results show that the improved IIM method has higher interpolation precision, which can consequently provide a valuable source of reference for FSI analysis by improving the precision of data transfer and satisfying the conservation of total force, total torque, and total virtual work in the meantime.

Isoparametric hypersurfaces in conic Finsler manifolds

In this paper, we introduce isoparametric functions and isoparametric hypersurfaces in conic Finsler spaces. We find that there are probably other isoparametric hypersurfaces in conic Minkowski spaces besides the conic Minkowski hyperplanes, conic Minkowski hyperspheres and conic Minkowski cylinders, such as helicoids. Moreover, we give a complete classification of isoparametric hypersurfaces in Kropina spaces with constant flag curvature.

Assessing the Accuracy of Different Remapping Methods in Adaptive Mesh Refinement

Additive manufacturing of metals has attracted much attention over the last years, promoting the development of several computational models for numerical simulation of the laser powder bed fusion (LPBF) process. Nevertheless, the finite element analysis of the LPBF process requires a large computational time. Thus, adaptive mesh refinement strategies are commonly adopted to reduce computational cost, which require some remapping procedure to transfer the state variables from the old mesh to the new one. The present study analyses two different remapping algorithms, namely the Inverse Isoparametric Mapping (IIM) and the Dual Kriging (DK) method. The IIM method uses the shape functions of the finite elements, while the DK method provides an explicit parametric interpolation. The case study adopted covers both coarsening and refinement procedures, using a mathematical function to define the mapped state variable. The accuracy of the remapping methods was lower in the refinement in comparison with the coarsening procedure. The error in the approximation is lower using the DK method in comparison with the IIM method. However, the IIM method does not suffer from error propagation in successive stages of either refinement/derefinement or coarsening/decoarsening.

Multiresolution Reconstruction of the Hypersonic Heat Flux

A multiresolution formulation for the direct and inverse reconstruction of the heat flux from temperature sensors distributed over a multidimensional solid in an hypersonic flow is presented. The thermal response is determined by approximating the system Green’s function with the Galerkin method and optimizing the heat flux distribution by fitting the distributed surface temperature data. Coating and glue layers are treated as separated domains for which the Green’s function is obtained independently. Connection conditions for the system Green’s function are derived by imposing continuity of heat flux and temperature concurrently at all interfaces. The heat flux is decomposed in a space-time basis where the temporal functions are multiresolution wavelet with arbitrary scaling. We develop quadrature formulas for the convolution product between wavelets and Green’s function, a reconstruction approach based on isoparametric mapping of three-dimensional geometries, and boundary wavelets for inverse problems. We validate the approach against turbulent conjugate heat transfer simulations and wind tunnel experiments at Mach 0.8 and 6. The main findings are that multidimensional effects are important near the wedge shoulder in the short time scale, that the L-curve regularization must be locally corrected to analyze transitional flows, and that proper regularization leads to sub-cell resolution of the inverse problem.

The Isoparametric Functions on a Class of Finsler Spheres

In this paper, we give global expressions of geodesics and isoparametric functions on a Randers sphere by navigation. We obtain isoparametric families and focal submanifolds in (S^{n}; F; d\mu_{BH}) by Cartan-M\"unzner polynomials. Further more, we construct some examples of closed and non-closed geodesics, isoparametric functions, isoparametric families and focal submanifolds.

Quadrilateral-area-coordinate-based numerical manifold method accommodating static and dynamic analysis

Quadrilateral mesh is a commonly used mathematical cover for numerical manifold method (NMM). However, the NMM based on quadrilateral isoparametric mapping has some intrinsic shortcomings, which is revealed for the first time in this study. Therefore, a new NMM is established to overcome these drawbacks by adopting a quadrilateral area coordinate system. In the proposed NMM, the stiffness matrix can be analytically determined without resorting to cumbersome Jacobian inversion and numerical integration. Moreover, a cone-complementary-based contact model is formulated in the context of the new NMM framework, which enables accurate determination of frictional and cohesive contact forces in solving dynamic contact problems. Thus, artificial penalty and open-close iteration in the original NMM can be all avoided. Several benchmark examples are simulated to demonstrate the excellent performance of the presented NMM.

Nodal integral method for 3D time-dependent anisotropic convection-diffusion equation

The main advantage of using Nodal Integral Methods (NIM) in solving partial differential equations (PDEs) is that they lead to an accurate solution over relatively coarse meshes. Due to the use of the transverse integration procedure in the derivation, most of the NIMs developed were restricted to PDEs with isotropic diffusion terms only. Recently, the NIM has been extended to arbitrary geometries using iso-parametric mapping approach, which resulted in the transformation of isotropic diffusion to anisotropic diffusion. This required the development of a method to approximate the cross-derivative terms that is consistent with the order of accuracy of traditional NIMs. In this paper, the 3D, time-dependent, anisotropic convection-diffusion equation is solved numerically using the NIM. A new method to approximate the cross-derivative terms based on the actual discrete unknowns of the NIM – namely, the line-averaged or surface-averaged variables – is developed. Also, the previously developed approximation for the cross-derivative terms in 2D (Kumar et al., 2013) that is based on the corner point values is extended to 3D. Six numerical test cases are solved to test the accuracy and efficiency of both approaches. The accuracy of NIM for the anisotropic diffusion equation is maintained even for coarse meshes for both cross-derivative approximations. However, the surface-averaged-based approximation is more accurate and efficient than the point-value-based approximation in all cases. The NIM using the surface-averaged-based approximation is second-order accurate in space and time. On the other hand, the NIM using the point-value-based approximation is between first and second-order accurate in space, and second-order accurate in time.

An ODE reduction method for the semi-Riemannian Yamabe problem on space forms

We consider the semi-Riemannian Yamabe type equations of the form −□u+λu=μ|u|p−1uonMwhere M is either the semi-Euclidean space or the pseudosphere of dimension m≥3, □ is the semi-Riemannian Laplacian in M, λ≥0, μ∈R∖︀{0} and p>1. Using semi-Riemannian isoparametric functions on M, we reduce the PDE into a generalized Emden–Fowler ODE of the form w′′+q(r)w′+λw=μ|w|p−1wonI,where I⊂R is [0,∞) or [0,π], q(r) blows-up at 0 and w is subject to the natural initial conditions w′(0)=0 in the first case and w′(0)=w′(π)=0 in the second. We prove the existence of blowing-up and globally defined solutions to this problem, both positive and sign-changing, inducing solutions to the semi-Riemannian Yamabe type problem with the same qualitative properties, with level and critical sets described in terms of semi-Riemannian isoparametric hypersurfaces and focal varieties. In particular, we prove the existence of sign-changing blowing-up solutions to the semi-Riemannian Yamabe problem in the pseudosphere having a prescribed number of nodal domains.

Some geometric correspondences for homothetic navigation

In this paper, we provide conceptional explanations for the geodesic and Jacobi field correspondences for homothetic navigation, and then let them guide us to the shortcuts to some well known flag curvature and S-curvature formulas. They also help us directly see the local correspondence between isoparametric functions or isoparametric hypersurfaces, which generalizes the classification works of Q. He and her coworkers for isoparametric hypersurfaces in Randers space forms and Funk spaces.