M Shadi Mohamed

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Verified

M Shadi verified their affiliation via an institutional email.

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• PhD, MSc, BSc
• Professor (Associate) at Heriot-Watt University

Problems of multiple scales of interest or of locally nonsmooth solutions may often involve heterogeneous media. These problems are usually very demanding in terms of computations with the conventional finite element method. On the other hand different enriched finite element methods such as the partition of unity which proved to be very successful...

An enriched partition of unity FEM is developed to solve time‐dependent diffusion problems. In the present formulation, multiple exponential functions describing the spatial and temporal diffusion decay are embedded in the finite element approximation space. The resulting enrichment is in the form of a local asymptotic expansion. Unlike previous wo...

Structural design of mechanical components is an iterative process that involves multiple stress analysis runs; this can be time consuming and expensive. It is becoming increasingly possible to make significant improvements in the efficiency of this process by increasing the level of interactivity. One approach is through real-time re-analysis of m...

In this paper, two high-order finite element models are investigated for the solution of two-dimensional wave problems governed by the Helmholtz equation. Plane wave enriched finite elements, developed in the Partition of Unity Finite Element Method (PUFEM), and high-order Lagrangian-polynomial based finite elements are considered. In the latter mo...

We investigate the inverse problem of identifying the wavenumber for the Helmholtz equation. The problem solution is based on measurements taken at few points from inside the computational domain or on its boundary. A novel iterative approach is proposed based on coupling the secant and the descent methods with the partition of unity method. Starti...

This paper proposes a novel dynamic response reconstruction method based on the Kalman filter which can simultaneously identifies external excitation and reconstructs dynamic responses at unmeasured positions. The weighted least squares method determines the load weighting matrix for excitation identification, while minimum variance unbiased estima...

We propose a class of high-order time integration schemes combined with high-order isogeometric analysis in three space dimensions. The combined methods offer robust solutions of nonlinear heat diffusion in three-dimensional composites that pose numerical challenges. This tailored strategy significantly enhances computational efficiency, especially...

Modeling and simulation have been extensively used to solve a wide range of problems in structural engineering. However, many simulations require significant computational resources, resulting in exponentially increasing computational time as the spatial and temporal scales of the models increase. This is particularly relevant as the demand for hig...

Inverse problems are studied in mathematics, science, and engineering, and they involve finding an unknown property of a medium or object from a probing excitation or observation. Inverse problems fit in with the Symmetry concept of this journal as they are the opposite of the associated forward problem, in which the causes are set and the effects...

In this paper, we present a new hybrid method that combines data and numerical simulations to address challenges associated with solving forward and inverse wave problems, specifically in the mid-frequency ranges. The computational demands of these problems can be overwhelming, even for relatively small computational domains. We propose a significa...

The present study proposes a novel approach for efficiently solving an anisotropic transient diffusion problem using an enriched finite element method. We develop directional enrichment for the finite elements in the spatial discretization and a fully implicit scheme for the temporal discretization of the governing equations. Within this comprehens...

We present a novel method for real-time fault classification using the time history of acoustic emissions (AEs) recorded from a lab-scale gas turbine operating under normal and faulty conditions across multiple turbine speeds. Time-frequency features are extracted using the continuous wavelet transform, and for each signal, the root mean square (RM...

A new adaptive finite volume method is proposed for the simulation of the wave problems in the time domain. The transient wave equations are discretized in time and space. A vertex-centered finite volume method is constructed with both cell-centered and edge-midpoint of each control volume. We then propose a mesh adaptation procedure based on energ...

In this paper, we propose a novel approach to solve nonlinear stress analysis problems in shell structures using an image processing technique. In general, such problems in design optimisation or virtual reality applications must be solved repetitively in a short period using direct methods such as nonlinear finite element analysis. Hence, obtainin...

The distributed dynamic load is difficult to obtain due to the complexity of loads in practical engineering, such as the aerodynamic loads of aircraft and the distributed dynamic loads of sea-crossing bridges. Thus, distributed dynamic load identification is important to deal with these difficulties, which is generally an ill-posed problem consider...

The implementation methods of finite element analysis (FEA) have remained essentially unchanged since the inception of FEA in the 1960s. Alterations of any of the input or design parameters to the FEA model can potentially nullify the previous results and subsequent additional simulations will be required. This is particularly relevant for situatio...

The inverse problem and the direct problem are symmetrical to each other. As a mathematical method for inverse problems, dynamic load identification is applicable to the situation when the load acting on the structure is difficult to measure directly. In addition, in most practical fields, the exact value of the structural parameters cannot be obta...

The paper proposes solving transient heat transfer in plates using high-order isogeometric analysis and high-order time integration schemes. The problem is often faced in fire-structure interaction where the heat transfer is coupled with the stress analysis. A major advantage for the proposed approach comes from high order continuity between elemen...

Many engineering structures are made of metal composite materials. External load information is a key issue for the design and condition monitoring of the structures. Due to the limitation of measurement technology and the external environment, it is difficult to directly measure dynamic loads on structures in many circumstances. This paper focuses...

In order to ensure the reliability of the structural design, it is necessary to know the external loads acting on the structure. In this paper, we propose a novel method to identify the dynamic loads based on function principles in the time domain. Assuming the external load remains constant within one micro segment, we establish a linear relations...

CALL FOR PAPERS: We are pleased to announce that the 9th International Conference on Modeling, Simulation and Applied Optimization will be held in Marrakesh, Morocco on April 26th-28th 2023 (www.icmsao.org/). ICMSAO provides a venue for engineers, mathematicians, and scientists from all over the world to share their latest research results in the f...

The modelling of fatigue using machine learning (ML) has been gaining traction in the engineering community. Among ML techniques, the use of probabi-listic neural networks (PNNs) has recently emerged as a candidate for modelling fatigue applications. In this paper, we use PNNs with nonconstant variance to model fatigue. We present two case studies...

Many experiments are usually needed to quantify probabilistic fatigue behavior in metals. Previous attempts used separate artificial neural network (ANN) to calculate different probabilistic ranges which can be computationally demanding for building probabilistic fatigue constant life diagram (CLD). Alternatively, we propose using probabilistic neu...

The risk of vibration-induced fatigue in process pipework is usually assessed through vibration measurements. For small-bore pipework, integrity personnel would measure the vibration of the pipework and refer to widely used charts to quantify the risk of vibration-induced fatigue. If the vibration levels are classified as OK, no action is required...

The determination of structural dynamic characteristics can be challenging, especially for complex cases. This can be a major impediment for dynamic load identification in many engineering applications. Hence, avoiding the need to find numerous solutions for structural dynamic characteristics can significantly simplify dynamic load identification....

Up until now, we have been using finite element analysis with problems modelled from two-noded elements. We will now see how a three-noded element, or quadratic element, will be used in finite element analysis. However, before we can look at the three-noded element, we must look at a few factors that will help in the analysis. We will start by disc...

Up until now, we have modelled and solved problems with finite element analysis using the actual coordinates of the structure. We can refer to these as the global coordinates.

As we are using local coordinates , the strain-displacement matrix, [B], and the Jacobian, [J], for the two dimensional element are functions of the ξ and η. Due to this, the integrands are no longer in the form of simple polynomials and are therefore difficult, if not impossible, to directly integrate these. However, it is possible to obtain an ap...

We can use virtual work to ensure that the internal work done by node 1, node 2, and node 3, is the equivalent to the external work done by the uniformly distributed load w, for any set of displacements. The displacement shape is given by the nodal displacements and the shape functions. Therefore, the displacement can be considered in three compone...

Full video of the lecture is available here: https://youtu.be/dZp9IfmWEsQ

Up until now, we have used two-noded elements to solve fi�nite element problems. We will now look at the three-noded element, also know as a quadratic element, and how it can be applied to finite element problems. Figure 4.1 represents an element with three nodes.

Up until now, we have looked at one dimensional problems with axially loaded beams. We have seen how we can apply the finite element method to these problems and we have seen how we can solve them. In this unit, we will now move into the next dimension and see how finite element analysis can be applied and used to solve two-dimensional problems.

A full video of the lecture is available here: https://youtu.be/0fwwlXSOR0o

Dynamic load identification is an inverse problem concerned with finding the load applied on a structure when the dynamic characteristics and the response of the structure are known. In engineering applications, some of the structure parameters such as the mass or the stiffness may be unknown and/or may change in time. In this paper, an online dyna...

In this paper we investigate the iterative solution of enriched finite element methods for solving a frequency domain wave problem. The considered methods are partition of unity isogeometric analysis (PUIGA) and the partition of unity finite element method (PUFEM). We study the performance of an operator based preconditioner, namely, the shifted La...

We propose a fast method for identifying the diffusion properties of a material based on few heat measurements taken at random parts of it. This is achieved by solving the inverse diffusion problem in order to evaluate the material heat diffusion coefficient. In the present study, we are interested in solving problems when the heat diffuses through...

Numerical solutions of wave problems are often influenced by uncertainties generated by a lack of knowledge of the input values related to the domain data and/or boundary conditions in the mathematical equations used in the modeling. Conventional methods for uncertainty quantification in modeling waves constitute severe challenges due to the high c...

A residual a-posteriori error estimate is used in conjunction with a q−adaptive procedure for selecting enrichment functions in the modelling of transient heat diffusion problems with the Generalized Finite Element Method (GFEM). The error estimate allows to assess local as well as global errors of the GFEM solutions and provides a tool to adaptive...

Evaluating dynamic loads in real time is crucial for health monitoring, fault diagnosis and fatigue analysis in aerospace, automotive and earthquake engineering among other vibration related applications. Developing such algorithms can be vital for several safety and performance functionalities. Therefore, over the past few years the identification...

We introduce the augmented Tikhonov regularization method motivated by Bayesian principle to improve the load identification accuracy in seriously ill-posed problems. Firstly, the Green kernel function of a structural dynamic response is established; then, the unknown external loads are identified. In order to reduce the identification error, the a...

Heat radiation in optically thick non-grey media can be well approximated with the Rosseland model which is a class of nonlinear diffusion equations with convective boundary conditions. The optical spectrum is divided into a set of finite bands with constant absorption coefficients but with variable Planckian diffusion coefficients. This simplifica...

Human drivers take instant decisions about their speed, acceleration and distance from other vehicles based on different factors including their estimate of the road roughness. Having an accurate algorithm for real-time evaluation of road roughness can be critical for autonomous vehicles in order to achieve safe driving and passengers comfort. In t...

We propose a high-order enriched partition of unity finite element method for linear and nonlinear time-dependent diffusion problems. The solution of this class of problems often exhibits non-smooth features such as steep gradients and boundary layers which can be very challenging to recover using the conventional low-order finite element methods....

This article investigates the effect of the selection of enrichment functions on the formulation of the Generalized Finite Element Method (GFEM) for the solutions of transient heat conduction problems. We present the study of an a-posteriori error estimate with the aim to show it as a reliable tool for the selection of enrichment functions to effic...

Predicting room acoustics using wave-based numerical methods has attracted great attention in recent years. Nevertheless, wave-based predictions are generally computationally expensive for room acoustics simulations because of the large dimensions of architectural spaces, the wide audible frequency ranges, the complex boundary conditions, and inher...

In this article, a study of residual based a posteriori error estimation is presented for the partition of unity finite element method (PUFEM) for three‐dimensional (3D) transient heat diffusion problems. The proposed error estimate is independent of the heuristically selected enrichment functions and provides a useful and reliable upper bound for...

Solving wave problems with isogeometric analysis has attracted a significant attention in the past few years. It is well known that keeping a fixed number of degrees of freedom per wavelength leads to an increased error as higher wavenumbers are considered. This behaviour often cited as the pollution error, improves significantly with isogeometric...

In this article, a study of residual based a-posteriori error estimation is presented for the partition of unity finite element method (PUFEM) for three-dimensional transient heat diffusion problems. The proposed error estimate is independent of the heuristically selected enrichment functions and provides useful and reliable upper bound for the dis...

We propose a non-intrusive stochastic model reduction method for polynomial chaos representation of acoustic problems using proper orthogonal decomposition. The random wavenumber in the well-established Helmholtz equation is approximated via the polynomial chaos expansion. Using conventional methods of polynomial chaos expansion for uncertainty qua...

We present a partition of unity finite element method for wave propagation problems in the time domain using an explicit time integration scheme. Plane wave enrichment functions are introduced at the finite elements nodes which allows for a coarse mesh at low order polynomial shape functions even at high wavenumbers. The initial condition is formul...

An efficient partition of unity finite element method for three-dimensional transient diffusion problems is presented. A class of multiple exponential functions independent of time variable is proposed to enrich the finite element approximations. As a consequence of this procedure, the associated matrix for the linear system is evaluated once at th...

The aim of this paper is to accurately solve short wave scattering problems governed by the Helmholtz equation using the Bernstein-Bézier Finite Element method (BBFEM), combined with a conformal perfectly matched layer (PML). Enhanced PMLs, where curved geometries are represented by means of the blending map method of Gordon and Hall, are numerical...

It is well known that Galerkin finite element methods suffer from pollution error when solving wave problems. To reduce the pollution impact on the solution different approaches were proposed to enrich the finite element method with wave-like functions so that the exact wavenumber is incorporated into the finite element approximation space. Solving...

The purpose of this work is to accurately solve short wave problems governed by the Helmholtz equation using Bernstein-Bézier Finite Elements method (BBFEM) [1] combined to conformal perfectly matched layers (PMLs). To ensure accurate boundary representation of curved edges and hence further enhance the PML, the linear blending function method due...

This work studies for the first time the solution of a nonlinear problem using an enriched finite element approach. Such problems can be highly demanding computationally. Hence, they can significantly benefit from the efficiency of the enriched finite elements. A robust partition of unity finite element method for solving transient nonlinear diffus...

In this work, the Bernstein-Bézier Finite Element Method (BBFEM) is implemented to solve short wave problems governed by the Helmholtz equation on unstructured triangular mesh grids. As for the hierarchical Finite Element (FE) approach, this high order FE method benefits from the use of static condensation which is an efficient tool for reducing th...

The Finite Element Method (FEM) has long established itself as an attractive choice for numerical modelling of wave scattering problems. The key feature of FEM is that it is a general numerical method that can be used for modelling complex geometries and heterogeneous materials. The NURBS based or Isogeometric FEM (IGAFEM) has been shown to be more...

Most of generalized finite element methods use dense direct solvers for the resulting linear systems. This is mainly the case due to the ill-conditioned linear systems that are associated with these methods. In the current study we investigate the performance of a class of iterative solvers for the generalized finite element solution of time-depend...

The method of using implicit time integration along with enriched finite element method has shown good potential as an approach to solve the wave equation in time[1]. Although robust and stable, the method inherently requires the solution of a system of linear equations every time step, which could be computationaly expensive especially if the doma...

This paper proposes a novel scheme for the solution of the electromagnetic wave equation in the time domain. A discretization scheme in time is implemented to render implicit solutions of systems of equations possible. The scheme allows for calculation of the field values at different time steps in an iterative fashion. The spatial grid is partitio...

We propose the study of a posteriori error estimates for time–dependent generalized finite element simulations of heat transfer problems. A residual estimate is shown to provide reliable and practically useful upper bounds for the numerical errors, independent of the heuristically chosen enrichment functions. Two sets of numerical experiments are p...

This paper investigates the accuracy of the partition of unity method (PUM) for the solution of time dependent heat transfer problems. We propose a mathematically rigorous, computable error estimate and test the accuracy of PUM against this error estimate. It is shown that the proposed error estimate provides reliable and practically useful upper b...

Computer aided design of mechanical components is an iterative process that often involves multiple stress analysis runs; this can be time consuming and expensive. Significant efficiency improvements can be made by increasing interactivity at the conceptual design stage. One approach is through real-time re-analysis of models with continuously upda...

This paper investigates an error estimate for the partition of unity method solution of time dependent diffusion problem. An enriched finite element method is used to solve the problem and the results are compared to a predefined error estimate. Gaussian functions with two different variations are used to compute the results and the one with better...

Radiative cooling in glass manufacturing is simulated using the partition of unity finite element method. The governing equations consist of a semi-linear transient heat equation for the temperature field and a stationary simplified P1 approximation for the radiation in non-grey semitransparent media. To integrate the coupled equations in time we c...

We investigate the effectiveness of the partition-of-unity finite element method for transient conduction-radiation problems in diffusive grey media. The governing equations consist of a semi-linear transient heat equation for the temperature field and a stationary diffusion approximation to the radiation in grey media. The coupled equations are in...

This paper presents a finite element method for the solution of Helmholtz problems at high wave numbers that offers the potential of capturing many wavelengths per nodal spacing. This is done by constructing oscillatory shape functions as the product of polynomial shape functions and either Bessel functions or planar waves. The resulting elementary...

The Partition of Unity Finite Element Method is used to solve wave scattering problems governed by the Helmholtz equation, involving one or more scatterers, in two dimensions. The method allows us to relax the traditional requirement of around ten nodal points per wavelength used in the Finite Element Method. Therefore the elements are multi-wavele...

The Partition of Unity Finite Element Method (PUFEM) is used to solve a wave scattering problem governed by the Helmholtz equation. The number q of the enriching plane waves is usually considered to be constant all over the computational domain which is a reasonable choice when a uniform mesh grid is considered. However, for nonuniform mesh grids w...