Finite Element Method Mesh Research Articles

Finite element reverse time migration imaging in tunnels using an unstructured mesh

Wave field extrapolation is critical in reverse time migration (RTM). At present, wavefield extrapolation in RTM imaging for tunnels is mostly carried out via the finite difference method. However, complex tunnel models, such as those for karst and fault fracture zones, are constructed using regular grids with straight curves which can easily cause numerical dispersion and reduce imaging accuracy. In this study, the wavefield extrapolation for tunnel RTM was conducted using the finite element method, where an unstructured mesh is employed as the body-fitted partition in a complex model. The Poynting vector calculation equation suitable for the unstructured mesh finite element method was established to suppress low-frequency noise interference. The tunnel space was considered during the wavefield extrapolation to suppress mirror artefacts by using the flexibility of mesh generation. Finally, the influence of the survey layouts (one-sidewall and two-sidewall) on the tunnel imaging results was explored. The RTM results for a simple tunnel model with an inclined interface show that the method based on unstructured meshes can effectively suppress low-frequency noise and mirror artefacts, thus obtaining clearer imaging results. Also, the two-sidewall tunnel survey layout more accurately obtains the real position of the inclined interface ahead of the tunnel face. Complex tunnel numerical modelling and actual data migration results further illustrate the effectiveness of the finite element unstructured mesh method.

Fracture propagation using the radial point interpolation method

AbstractThis work presents a crack path prediction algorithm combined with the radial point interpolation method (RPIM), a meshless method. To allow easier implementation in existing structural analysis software, this algorithm is numerically compatible with finite element method (FEM) formulations. The proposed RPIM formulation uses the triangular elements of a FEM mesh as the background integration grid, allowing also to combine both formulations more easily. Thus, with the developed methodology, the RPIM can be integrated directly in a FEM software or use the same CAD tools to build the discretization meshes. Because the RPIM shape functions possess the delta Kronecker property, all the numerical techniques available for the FEM can be applied to enforce the natural and essential boundary conditions.The developed algorithm uses the maximum tangential stress criterion to determine the crack propagation direction, and the crack paths obtained with it corresponded well to previous research.

Noise-robust bioimpedance approach for cardiac output measurement

Objective: Congestive heart failure is a problem affecting millions of Americans. A continuous, non-invasive, telemonitoring device that can accurately monitor cardiac metrics could greatly help this population, reducing unnecessary hospitalizations and cost. Approach: Machine learning (ML) algorithms trained on electrical-impedance tomography (EIT) data are presented for portable cardiac monitoring. The approach was validated on a simulated thorax and a measured tank experiment. A highly detailed 4D chest model (finite element method mesh and conductivity profiles) was developed utilizing the 4D XCAT phantom to provide realistic data. The ML algorithms were trained using databases that assumed the presence of poorly contacting electrodes without any assumptions of knowing which electrodes would be bad in the experiment. The trained ML algorithms were compared to EIT evaluated with and without removing bad electrodes. Main results: A regression support vector machine and a deep neural network (DNN) were found to be the most accurate and robust to poorly contacting electrodes while not needing to know which electrodes were in poor contact in the simulated and measured experiments, respectively. Significance: Although the ML algorithms are not always better than EIT (with bad electrodes removed), the comparable results without needing a priori knowledge of which electrodes are bad is seen as a very promising feature. An evaluation of computational costs showed that the DNN required comparable computational power to the other methods while requiring less memory, which could make the DNNs an attractive algorithm for a low-power, portable system. This work represents an important validation of the method using measured data, and model development, which is needed to apply this method on real clinical data. Additionally, the developed 4D simulated thorax model could be an important tool within the EIT community.

Comparison of weak and strong formulations for 3D stress predictions of composite beam structures

Accurate full field stress responses are often necessary to predict the structural performance of composite structures. The Unified Formulation (UF) is a promising approach to realise efficient and accurate 3D stress predictions of beam-like structures by using recently proposed hierarchical Serendipity Lagrange Elements (SLE) for capturing the 2D response of the beam cross-section while the 1D behaviour along the beam axis is captured using the Finite Element Method (FEM). Despite the computational merits of SLE elements, the performance of UF-SLE-FEM model is strongly influenced by FEM mesh discretisation along the beam's longitudinal axis. With respect to multi-layered beam structures, a high density FEM mesh may lead to loss of efficiency of the UF-SLE-FEM model due to a significant increase in the number of degrees of freedom required to obtain convergence. This study proposes high-order refined formulations of the 1D structure based on strong-form, Differential Quadrature Method (DQM) and weak form, FEM for 3D stress analysis of composite structures. The proposed high-order strong-form DQM and weak-form FEM models, which are benchmarked against exact solutions, lead to significant computational advantages over UF-SLE-FEM models of similar accuracies. However, the symmetry and positive definiteness of the stiffness matrices of FEM models offer a numerical advantage over the strong-form DQM model with non-symmetric, non-positive definite stiffness matrices.

The inverse problem and the second order [formula omitted] scheme with finite element method used for 2D nonlinear space fractional Schrödinger equation

In this paper, we do research on the numerical analysis and the inverse problem for the two-dimensional nonlinear space fractional Schrödinger equation. The second order θ scheme combined with the unstructured mesh finite element method, which can cover the second order Crank–Nicolson method and the second order backward difference method in time, and can deal with the two-dimensional irregular region boundaries better in space, is used to obtain the numerical solution of the considered equation. On this basis, we introduce the Bayesian method to inverse the unknown parameters for the two-dimensional fractional Schrödinger equation. To testify the validity and efficiency of the proposed methods, a numerical example is conducted.

An adjustable degrees-of-freedom numerical method for computing the temperature distribution of electrical devices

A novel adjustable degrees-of-freedom numerical method for computing the temperature distribution of electrical devices is presented. The proposed method provides a convenient, accurate and efficient approach to increase/reduce variables and couple/decouple variables. In traditional finite element method (FEM), the electromagnetic field and thermal field need different FEM meshes, which reduce the efficiency and accuracy of the field coupling. Also, the traditional method introduces the inevitable error when the power loss is plotted from Maxwell to Fluent as the different meshes are employed. In this paper, the greatest advantage is that the electromagnetic field and thermal field can be computed using a single mesh with the proposed technology of exploiting different constrained degrees-of-freedom, which decreases the computation time and system complexity. The global domain divided into several sub-domains were utilizing various corresponding functions and also employed to reduce the computational burden when calculating the core loss, copper loss, stray loss and temperature distribution. Numerical examples are reported to showcase the accuracy and effectiveness of the proposed method.

Modeling and simulation of hypothermia effects on cardiac electrical dynamics.

Previous experimental evidence has shown the effect of temperature on the action potential duration (APD). It has also been demonstrated that regional cooling of the heart can prolong the APD and promote the termination of ventricular tachycardia. The aim of this study is to demonstrate the effect of hypothermia in suppressing cardiac arrhythmias using numerical modeling. For this purpose, we developed a mathematical model that couples Pennes’ bioheat equation and the bidomain model to simulate the effect of heat on the cardiac action potential. The simplification of the proposed heat–bidomain model to the heat–monodomain model is provided. A suitable numerical scheme for this coupling, based on a time adaptive mesh finite element method, is also presented. First, we performed two-dimensional numerical simulations to study the effect of heat on a regular electrophysiological wave, with the comparison of the calculated and experimental values of Q10. Then, we demonstrated the effect of global hypothermia in suppressing single and multiple spiral waves.

Stiffness design of plate/shell structures by evolutionary topology optimization

This work aims at developing a simple and viable evolutionary design approach to optimally orient stiffeners for plate and shell structures so as to maximize stiffness while satisfying volume requirements. The evolutionary algorithm is transformed into a mathematical model, where stiffeners are treated as being alive, and the optimization is perceived as an adaptive growth procedure that starts from the constraint points and extends along the gradient directions of plate and shell stiffness. To eliminate the expensive re-meshing upon design changes, a special numerical treatment called “stiffness transformation approach” is developed. In this approach, the stiffness matrix of growing stiffeners is interpolated within their surrounding regions, and the stiffness of neighboring finite elements is modified to simulate the presence of stiffeners. Such a transformation allows the growing stiffeners to be mathematically separated from the underlying finite element method mesh; thus, stiffeners can extend toward arbitrary directions to form an optimized layout solution. An easy-to-use implementation of the evolutionary algorithm is demonstrated in detail through a machine tool design example. Compared to the original design, both numerical and experimental tests confirm that the stiffness in the proposed design is improved; as a result, new possibilities emerge for the design of large-scale plate and shell structures in engineering.

An adaptive degrees-of-freedom finite element method for 3-D nonlinear magneto-thermal field analysis

An adaptive degrees-of-freedom finite element method (FEM) for 3-dimensional nonlinear magneto-thermal fields is proposed in this article. Both magnetic field and thermal field are discretized using a single FEM mesh, and their degrees of freedom (DoFs) are individually controlled based on the field characteristics. Before solving the algebraic equations, the constrained DoFs are removed using the slave-master technique. The computing time and the storage resources of the second set of FEM mesh are saved while meeting the different requirements on discretization. Data transfer between the thermal and magnetic fields are easily implemented, and mapping errors between different meshes are avoided. To showcase the feasibility and the effectiveness of this method, several numerical examples are tested.

An unstructured mesh finite element method for solving the multi-term time fractional and Riesz space distributed-order wave equation on an irregular convex domain

• A multi-term time fractional and Riesz space distributed-order wave equation is discussed. • Firstly the equation is transformed into a multi-term time-space fractional wave equation. • The equation is solved by discretising using C–N scheme and the FEM with an unstructured mesh. • Stability and convergence are investigated. • Some examples are provided to show the effectiveness and correctness of the proposed numerical method. In this paper, the numerical analysis for a multi-term time fracstional and Riesz space distributed-order wave equation is discussed on an irregular convex domain . Firstly, the equation is transformed into a multi-term time-space fractional wave equation using the mid-point quadrature rule to approximate the distributed-order Riesz space derivative. Next, the equation is solved by discretising in time using a Crank–Nicolson scheme and in space using the finite element method (FEM) with an unstructured mesh, respectively. Furthermore, stability and convergence are investigated by introducing some important lemmas on irregular convex domain. Finally, some examples are provided to show the effectiveness and correctness of the proposed numerical method.

Modeling hydraulic fracture in heterogeneous rock materials using permeability-based hydraulic fracture model

Hydraulic fracturing is one of the most important techniques for enhancing oil/gas production. The permeability-based hydraulic fracture (PHF) model, which is based on the smeared-crack method and considers the interaction between the pore pressure and solid phase, is adopted in the present study for a fully-coupled simulation of the hydraulic fracture in a heterogeneous rock formation. The level set method (LSM), which is used to describe the distribution of material properties of heterogeneous rocks, is coupled with the PHF model. Using the coupled PHF–LSM model, a series of finite element method (FEM) simulations are carried out to investigate the characteristics of a hydraulic fracture (e.g., the breakdown pressure and fracture propagation) in heterogeneous rocks. Three types of heterogeneous rocks are examined: layered rock, rock with distributed inclusions, and rock with random spatial variations in the material properties. The results of the numerical simulations show that the coupled PHF–LSM model can describe the material interface without changing the FEM mesh used to discretize the physical domain. Further, the model effectively simulates hydraulic-fracturing problems for various heterogeneous rocks.

AN EXPERIMENTAL ANALYSIS AND VALIDATION OF ELECTRICAL IMPEDANCE TOMOGRAPHY TECHNIQUE FOR MEDICAL OR INDUSTRIAL APPLICATION

Electrical impedance tomography is a recently established technique by which impedance of an object (medical or nonmedical applications) is measured data from the surface of the object, and a numerically simulated reconstruction of the object internal shape of the image can be obtained. This imaging technique based on boundary or surface voltage is measured when the different current pattern is injected into it. For current pulse, we are creating a voltage controlled current source, which is based on the different RC circuits, according to current amplitude and frequency values. The current source used in inject the current pulse of the various phantoms. The current position and measuring voltage is controlled by the created control unit or programmable system on chip (PSOC) of the proposed EIT system. After that image reconstruction of the cross-sectional image of resistivity requires sufficient data collection from used phantoms, which is based on finite element method (FEM) method and Tikhonov regularization method with helps of graphical user interface (GUI) on MatLab. The objective of the GUI was to produce an image (2D/3D), impedance distribution graph, and the FEM mesh model according to used electrode combinations from the various phantoms. EIT system has a great potential for imaging modality, is non-invasive, radiation-free, and inexpensive for medical applications.

A Positivity Preserving Moving Mesh Finite Element Method for the Keller–Segel Chemotaxis Model

In this paper, we present an efficient adaptive moving mesh finite element method for the numerical solution of the Keller–Segel chemotaxis model. The mesh points are continuously redistributed by a coordinate transformation defined from the computational domain to the physical domain to concentrate the grid nodes in regions of large solution variations in the physical domain. The Keller–Segel equations are discretized using an implicit-explicit finite element method using piecewise polynomials defined on triangular meshes. The spatial discretization scheme is designed with a positivity preserving property, if the initial solutions of the physical model are positive then the computed solutions stay positive at all time levels. Several numerical experiments are presented to demonstrate the performance of the proposed method for solving Keller–Segel model. The numerical results show that the proposed method can reduce the computational cost while improving the overall accuracy of the computed solutions of the Keller–Segel model.

Simulating damage evolution and fracture propagation in sandstone containing a preexisting 3‐D surface flaw under uniaxial compression

SummaryPreexisting flaws and rock heterogeneity have important ramifications on the process of rock fracturing and on rock stability in many applications. Therefore, there is great interest in numerical modelling of rock fracture and the underlying mechanisms. We simulated damage evolution and fracture propagation in sandstone specimens containing a preexisting 3‐D surface flaw under uniaxial compression. We applied the linear elastic damage model based on the unified strength theory following the rock failure process analysis code. However, in contrast to the rock failure process analysis code, we used the finite element method with tetrahedron elements on unstructured meshes. It provided higher geometrical flexibility and allowed for a more accurate representation of the disk‐shaped flaw with various flaw depths, angles, and lengths through locally adapted meshes. The rock heterogeneity was modelled by sampling the initial local Young's modulus from a Weibull distribution over a cubic grid. The values were then interpolated to the computational finite element method mesh. This method introduced an additional length scale for the rock heterogeneity represented by the cell size in the sampling grid. The generation of three typical surface cracking patterns, called wing cracks, anti‐wing cracks, and far‐field cracks, were identified in the simulation results. These depend on the geometry of the preexisting surface flaw. The simulated fracture propagation, coalescence types, and failure modes for the specimens with preexisting surface flaw show good agreement with recent experimental studies.

Numerical identification of the thermal conductivity tensor and the heat capacity per unit volume of an anisotropic material

In this paper, an inverse approach based on gradient conjugate method for thermal conductivity tensor and heat capacity per unit volume measurement is summarized. A suitable analysis is done for the mesh in finite element method and for the time steps for the time integration. For a composite material, it is shown the importance to identify the thermal conductivity tensor components in the principal axes.

Natural Frequencies of Submerged Structures Using an Efficient Calculation of the Added Mass Matrix in the Boundary Element Method

This work presents an efficient way to calculate the added mass matrix, which allows solving for natural frequencies and modes of solids vibrating in an inviscid and infinite fluid. The finite element method (FEM) is used to compute the vibration spectrum of a dry structure, then the boundary element method (BEM) is applied to compute the pressure modes needed to determine the added mass matrix that represents the fluid. The BEM requires numerical integration which results in a large computational cost. In this work, a reduction of the computational cost was achieved by computing the values of the pressure modes with the required numerical integration using a coarse BEM mesh, and then, interpolation was used to compute the pressure modes at the nodes of a fine FEM mesh. The added mass matrix was then computed and added to the original mass matrix of the generalized eigenvalue problem to determine the wetted natural frequencies. Computational cost was minimized using a reduced eigenvalue problem of size equal to the requested number of natural frequencies. The results show that the error of the natural frequencies using the procedure in this work is between 2% and 5% with 87% reduction of the computational time. The motivation of this work is to study the vibration of marine mammals' ear bones.

Computer Algebra Challenges in Nanotechnology: Accurate Modeling of Nanoscale Electro-optic Devices Using Finite Elements Method

The simulation of silicon-based light-emitting and photodetectors nanodevices using computer algebra became a challenge. These devices couple the hyperbolic equations of electromagnetic radiation, the parabolic equations of heat conduction, the elliptic equations describing electric potential, and the eigenvalue equations of quantum mechanics—with the nonlinear drift–diffusion equations of the semiconductor physics. These complex equations must be solved by using generally mixed Dirichlet–Neumann boundary conditions in three-dimensional geometries. Comsol Multiphysics modeling software is employed integrated with MATLAB–SIMULINK and Zemax. The physical equations are discretized on a mesh using the Galerkin finite element method (FEM) and to a lesser extent the method of finite volumes. The equations can be implemented in a variety of forms such as directly as a partial differential equation, or as a variational integral, the so-called weak form. Boundary conditions may also be imposed directly or using variational constraint and reaction forces. Both choices have implication for convergence and physicality of the solution. The mesh is assembled from triangular or quadrilateral elements in two-dimensions, and hexahedral or prismatic elements in three dimensions, using a variety of algorithms. Solution is achieved using direct or iterative linear solvers and nonlinear solvers. The former are based on conjugate gradients, the latter generally on Newton–Raphson iterations. The general framework of FEM discretization, meshing and solver algorithms will be presented together with techniques for dealing with challenges such as multiple time scales, shocks and nonconvergence; these include load ramping, segregated iterations, and adaptive meshing.

An efficient finite element method for the two-dimensional nonlinear time–space fractional Schrödinger equation on an irregular convex domain

In this paper, we introduce an efficient Galerkin finite element method for solving the two-dimensional nonlinear time–space fractional Schrödinger equation, with time-dependent potential function, defined on an irregular convex domain. The Caputo time fractional derivative is discretized by the finite difference scheme, while the space fractional derivative defined on the irregular convex domain is dealt with the unstructured mesh finite element method. Given the nonlinear property of the considered equation, an efficient linearized strategy is used. To testify the efficiency of the proposed numerical scheme, two numerical examples are conducted with error and convergence analysis.

Lattice Boltzmann Simulation of Shallow Water Equations in Finite Element Framework

Lee, H. and Lee, N., 2018. Lattice Boltzmann Simulation of Shallow Water Equations in Finite Element Framework. In: Shim, J.-S.; Chun, I., and Lim, H.S. (eds.), Proceedings from the International Coastal Symposium (ICS) 2018 (Busan, Republic of Korea). Journal of Coastal Research, Special Issue No. 85, pp. 106–110. Coconut Creek (Florida), ISSN 0749-0208.The characteristic Galerkin finite element model for the discrete Boltzmann equation is developed for the depth-integrated shallow water equations. The lattice Boltzmann method (LBM) simulates microscopic collision and streaming of fluid particles and, then, evaluates the macroscopic velocity, pressure, etc. from the computed particle distribution. Due to the inherent geometric flexibility of the unstructured mesh in finite element method, the mesh clustering near large gradients and the numerical treatments of complex geometries has become possible. Some benchmark problems of practical importance (subcritical flow over hump, flow around cylinder, curved channel, etc.) are tested and the numerical results are in good agreement with previous experimental or analytic solutions.

Intensity of singular stress fields of an embedded fibre under pull-out force

Previous experimental studies of fibre pull-out test show two dangerous points on the interface. Failure usually occurs at the bonded end of the fibre (Point A) or at the entry point on the surface of the matrix (Point E). Both points have different singular stress fields which causes crack initiation, crack propagation, and final failure. In this paper, intensity of singular stress fields (ISSF) at the fibre bonded end A and ISSF at the intersection point E of the fibre and the surface are discussed. The analysis method focuses on calculating the finite element method (FEM) stress radio by using a reference model and an unknown model. In the unknown model and the reference model, same FEM mesh pattern is applied. To analyse the ISSF at A, the body force method solution is used as the reference model. To analyse the ISSF at E, the reciprocal work contour integral method (RWCIM) solution is used as the reference model. Then, the two ISSFs are compared and discussed by varying the fibre embedded length lin. When lin is shorter, the singular stress at A is larger than the singular stress at E. When lin is longer, the singular stress at E is larger than the ISSF at A.