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We construct a nonconforming virtual element method for the approximation of singular solutions to isotropic linear elasticity problems on polygonal domains. Standard nonconforming virtual element spaces are enriched with suitable singular functions. The enrichment is based on the nonconforming structure of the discrete spaces and not on partition of unity techniques. We prove optimal convergence and assess numerically the theoretical results of the method. The proposed scheme naturally paves the way for an efficient linear elastic fracture solver.

The Virtual Element Method (VEM) for the elasticity problem is considered in the framework of the Hu-Washizu variational formulation. In particular, a couple of low-order schemes presented in [1], are studied for quadrilateral meshes. The methods under consideration avoid the need of the stabilization term typical of the VEM, due to the introduction of a suitable projection on higher-order polynomials. The schemes are proved to be stable and optimally convergent in a compressible regime, including the case where highly distorted (even non-convex) meshes are employed.

There is an increasing demand for using deep underground space at various scales, such as small-scale geothermal wells and large-scale projects like tunnels or nuclear waste disposal. The deep underground space in fractured rock is a heterogeneous and challenging medium. Fractures have a significant impact on both the groundwater flow and the mechanical behavior. This thesis aims to develop analytic element models that capture the behavior of fractured rock for both groundwater flow and linear elasticity at different scales. Because these models are analytic in their formulation, they can model with machine precision and investigate behavior near singular points. For groundwater flow, the thesis deals with two approaches to capture the groundwater flow behavior in fractured rock: a continuum approach and a discrete fracture network approach. The continuum approach considers the impact of fractures by varying the hydraulic conductivity based on depth. This method allows for efficient modeling while still capturing the depth-dependent behavior accurately. The discrete approach, in turn, implements the fractures directly by embedding them in the continuum. Unlike previous models, this model implements intersection without the need for approximations. The discrete model also demonstrates how fractures with discontinuous transmissivity are connected to model a heterogeneous fracture network. For linear elasticity, the fractures and tunnels are modeled discretely in a plane strain continuum as analytic elements. These elements possess degrees of freedom, and no theoretical limitation exists on the number of elements. The execution of a model with 10,000 fractures effectively demonstrates the speed and accuracy of this method. Integrating seepage forces into the linear elastic model has improved the correlation between groundwater flow and linear elasticity. This enhancement allows for a more precise analysis of the impact of seepage forces near singular points. The soluti

There is an increasing demand for using deep underground space at various scales, such as small-scale geothermal wells and large-scale projects like tunnels or nuclear waste disposal. The deep underground space in fractured rock is a heterogeneous and challenging medium. Fractures have a significant impact on both the groundwater flow and the mechanical behavior. This thesis aims to develop analytic element models that capture the behavior of fractured rock for both groundwater flow and linear elasticity at different scales. Because these models are analytic in their formulation, they can model with machine precision and investigate behavior near singular points. For groundwater flow, the thesis deals with two approaches to capture the groundwater flow behavior in fractured rock: a continuum approach and a discrete fracture network approach. The continuum approach considers the impact of fractures by varying the hydraulic conductivity based on depth. This method allows for efficient modeling while still capturing the depth-dependent behavior accurately. The discrete approach, in turn, implements the fractures directly by embedding them in the continuum. Unlike previous models, this model implements intersection without the need for approximations. The discrete model also demonstrates how fractures with discontinuous transmissivity are connected to model a heterogeneous fracture network. For linear elasticity, the fractures and tunnels are modeled discretely in a plane strain continuum as analytic elements. These elements possess degrees of freedom, and no theoretical limitation exists on the number of elements. The execution of a model with 10,000 fractures effectively demonstrates the speed and accuracy of this method. Integrating seepage forces into the linear elastic model has improved the correlation between groundwater flow and linear elasticity. This enhancement allows for a more precise analysis of the impact of seepage forces near singular points. The soluti

Consider a crack propagating in a plane according to a loading that results in anti-plane shear deformation. We show here that if the crack path consists of two straight segments making an angle psi not equal 0 at their junction, examples can be provided for which the value of the stress-intensity factor (SIF) actually depends on the previous history of the motion. This is in sharp contrast with the rectilinear case (corresponding to psi = 0), where the SIF is known to have a local character, its value depending only on the position and velocity of the crack tip at any given time., Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

Consider a crack propagating in a plane according to a loading that results in anti-plane shear deformation. We show here that if the crack path consists of two straight segments making an angle psi not equal 0 at their junction, examples can be provided for which the value of the stress-intensity factor (SIF) actually depends on the previous history of the motion. This is in sharp contrast with the rectilinear case (corresponding to psi = 0), where the SIF is known to have a local character, its value depending only on the position and velocity of the crack tip at any given time., Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

There is an increasing demand for using deep underground space at various scales, such as small-scale geothermal wells and large-scale projects like tunnels or nuclear waste disposal. The deep underground space in fractured rock is a heterogeneous and challenging medium. Fractures have a significant impact on both the groundwater flow and the mechanical behavior. This thesis aims to develop analytic element models that capture the behavior of fractured rock for both groundwater flow and linear elasticity at different scales. Because these models are analytic in their formulation, they can model with machine precision and investigate behavior near singular points. For groundwater flow, the thesis deals with two approaches to capture the groundwater flow behavior in fractured rock: a continuum approach and a discrete fracture network approach. The continuum approach considers the impact of fractures by varying the hydraulic conductivity based on depth. This method allows for efficient modeling while still capturing the depth-dependent behavior accurately. The discrete approach, in turn, implements the fractures directly by embedding them in the continuum. Unlike previous models, this model implements intersection without the need for approximations. The discrete model also demonstrates how fractures with discontinuous transmissivity are connected to model a heterogeneous fracture network. For linear elasticity, the fractures and tunnels are modeled discretely in a plane strain continuum as analytic elements. These elements possess degrees of freedom, and no theoretical limitation exists on the number of elements. The execution of a model with 10,000 fractures effectively demonstrates the speed and accuracy of this method. Integrating seepage forces into the linear elastic model has improved the correlation between groundwater flow and linear elasticity. This enhancement allows for a more precise analysis of the impact of seepage forces near singular points. The soluti

There is an increasing demand for using deep underground space at various scales, such as small-scale geothermal wells and large-scale projects like tunnels or nuclear waste disposal. The deep underground space in fractured rock is a heterogeneous and challenging medium. Fractures have a significant impact on both the groundwater flow and the mechanical behavior. This thesis aims to develop analytic element models that capture the behavior of fractured rock for both groundwater flow and linear elasticity at different scales. Because these models are analytic in their formulation, they can model with machine precision and investigate behavior near singular points. For groundwater flow, the thesis deals with two approaches to capture the groundwater flow behavior in fractured rock: a continuum approach and a discrete fracture network approach. The continuum approach considers the impact of fractures by varying the hydraulic conductivity based on depth. This method allows for efficient modeling while still capturing the depth-dependent behavior accurately. The discrete approach, in turn, implements the fractures directly by embedding them in the continuum. Unlike previous models, this model implements intersection without the need for approximations. The discrete model also demonstrates how fractures with discontinuous transmissivity are connected to model a heterogeneous fracture network. For linear elasticity, the fractures and tunnels are modeled discretely in a plane strain continuum as analytic elements. These elements possess degrees of freedom, and no theoretical limitation exists on the number of elements. The execution of a model with 10,000 fractures effectively demonstrates the speed and accuracy of this method. Integrating seepage forces into the linear elastic model has improved the correlation between groundwater flow and linear elasticity. This enhancement allows for a more precise analysis of the impact of seepage forces near singular points. The soluti

One of the novel methodologies in computational physics research is to use mimetic discretisation techniques. Among these, the mimetic spectral element method holds special promise as it not only has the benefits of mimetic methods but also the additional benefit of higher-order discretisations using higher polynomial degrees. These methods are aided by the development of algebraic dual polynomials, resulting in a sparser system for better computational efficiency. This combination was used to develop a formulation that would result in topological relations for equilibrium of forces as well as the symmetry of the stress tensor for linear elasticity as well as the first steps for Stokes flows in an orthogonal domain. As a result, this study was extended to look at how a modified formulation would behave for an unsteady linear elastic solid, with the intention to extend this method to Fluid-Structure Interaction cases. However, the choice of both primal and nodal basis functions makes it impossible to undertake this challenge, demanding a rethink in strategy towards looking at linear elastic solids when the physical domain is not orthogonal. With the use of bundle-valued forms to represent physical quantities, a new hybrid formulation is developed where the equivalent of the physical problem is computed on a reference domain, which is orthogonal and thus can utilise the spectral bases defined before. The physical problem is defined on a skewed domain, where partial transformation of components results in a formulation that can conserve linear momentum point-wise, but not conservation of angular momentum, although angular momentum does converge on refinement of polynomial degree and mesh parameters. A change in bases with partial transformation aiming to make angular momentum conservation topological is not fruitful, although the value of the error decreases in the process. The final attempt is through full transformation, which results in a formulation with an inheren, Aerospace Engineering

The Hierarchical Subspace Iteration Method is a novel method used to compute eigenpairs of the Laplace-Beltrami problem. It reduces the number of iterations required for convergence by restricting the problem to a smaller space and prolonging the solution as a starting point. This method has shown great performance improvements for Laplace-Beltrami eigenproblems. We propose an adaptation to the Hierarchical Subspace Iteration Method that allows for computing vibration modes of elastic objects. We evaluate potential optimizations that can be made, as well as the performance characteristics of the method. Our method was shown to be faster than SIM in most cases while even beating Matlab's Lanczos solver in some cases., Computer Science

In this paper, we propose a simple energy decaying iterative thresholding algorithm to solve the two-phase minimum compliance problem. The material domain is implicitly represented by its characteristic function, and the problem is formulated into a minimization problem by the principle of minimum complementary energy. We prove that the energy is decreasing in each iteration. Two effective continuation schemes are proposed to avoid trapping into the local minimum. Numerical results on 2D isotropic linear material demonstrate the effectiveness of the proposed methods.

An original, variational formulation of the Virtual Element Method (VEM) is proposed, based on a Hu–Washizu mixed variational statement for 2D linear elastostatics. The proposed variational framework appears to be ideal for the formulation of VEs, whereby compatibility is enforced in a weak sense and the strain model can be prescribed a priori, independently of the unknown displacement model. It is shown how the ensuing freedom in the definition of the strain model can be conveniently exploited for the formulation of self-stabilized and possibly locking-free low order VEs. The superior performances of the VEs formulated within this framework has been verified by application to several numerical tests.

We provide an approximation result for the pure traction problem of linearized elasticity in terms of local minimizers of finite elasticity, under the constraint of vanishing average curl for admissible deformation maps. When suitable rotations are included in the constraint, the limit is shown to be the linear elastic equilibrium associated to rotated loads.

Computing the total energy of a system of N interacting dislocations in the presence of arbitrary free surfaces is a difficult task, requiring Finite Element (FE) numerical calculations. Worst, high accuracy requires very fine meshes in the proximity of each dislocation core. Here we show that FE calculations can be conveniently replaced by a Machine Learning (ML) approach. After formulating the elastic problem in terms of one and two-body terms only, we use Sobolev training to obtain consistent information on both energy and forces, fitted using a feed-forward neural network (NN) architecture. As an example, we apply the proposed methodology to corrugated, heteroepitaxial semiconductor films, searching for the minimum-energy dislocation distributions by using Monte Carlo. Importantly, the presence of an interaction cutoff allows for the application of the method to systems of different sizes without the need to repeat training. Millions of energy evaluations are performed, a task which would have been impossible by brute-force FE calculations. Finally, we show how forces can be exploited in running 2D ML-based dislocation dynamics simulations.

We present an analytic formulation for an elastic medium under uniform stress and in plane strain, with a very large number of cracks that are allowed to intersect. The singularities at the tips of the cracks are represented exactly; the solution is written in terms of series expansions of negative powers of a complex variable defined outside the unit circle in a complex reference plane. We modeled an assembly of 10,000 cracks and show stress trajectories both for the whole model and for individual cracks, chosen at random from the assembly.

We present an analytic formulation for an elastic medium under uniform stress and in plane strain, with a very large number of cracks that are allowed to intersect. The singularities at the tips of the cracks are represented exactly; the solution is written in terms of series expansions of negative powers of a complex variable defined outside the unit circle in a complex reference plane. We modeled an assembly of 10,000 cracks and show stress trajectories both for the whole model and for individual cracks, chosen at random from the assembly.

We present an analytic formulation for an elastic medium under uniform stress and in plane strain, with a very large number of cracks that are allowed to intersect. The singularities at the tips of the cracks are represented exactly; the solution is written in terms of series expansions of negative powers of a complex variable defined outside the unit circle in a complex reference plane. We modeled an assembly of 10,000 cracks and show stress trajectories both for the whole model and for individual cracks, chosen at random from the assembly.

Computer Aided Design (CAD) software packages are used in the industry to design mechanical systems. Then, calculations are often performed using simulation software packages to improve the quality of the design. To speed up the development costs, companies and research centers have been trying to ease the integration of the computation phase in the design phase. The collocation methods have the potential of easing such integration thanks to their meshless nature. The geometry discretization step which is a key element of all computational method is simplified compared to mesh-based methods such as the finite element method. We propose in this thesis a unified workflow that allows the solution of engineering problems defined by partial differential equations (PDEs) directly from input CAD files. The scheme is based on point collocation methods and proposed techniques to enhance the solution. We introduce the idea of “smart clouds”. Smart clouds refer to point cloud discretizations that are aware of the exact CAD geometry, appropriate to solve a defined problem using a point collocation method and that contain information used to improve locally the solution. We introduce a unified node selection algorithm based on a generalization of the visibility criterion. The proposed algorithm leads to a significant reduction of the error for concave problems and does not have any drawback for convex problems. The point collocation methods rely on many parameters. We select in this thesis parameters for the Generalized Finite Difference (GFD) method and the Discretization-Corrected Particle Strength Exchange (DC PSE) method that we deem appropriate for most problems from the field of linear elasticity. We also show that solution improvement techniques, based on the use of Voronoi diagrams or on a stabilization of the PDE, do not lead to a reduction of the error for all of the considered benchmark problems. These methods shall therefore be used with care. We propose two types of

Computer Aided Design (CAD) software packages are used in the industry to design mechanical systems. Then, calculations are often performed using simulation software packages to improve the quality of the design. To speed up the development costs, companies and research centers have been trying to ease the integration of the computation phase in the design phase. The collocation methods have the potential of easing such integration thanks to their meshless nature. The geometry discretization step which is a key element of all computational method is simplified compared to mesh-based methods such as the finite element method. We propose in this thesis a unified workflow that allows the solution of engineering problems defined by partial differential equations (PDEs) directly from input CAD files. The scheme is based on point collocation methods and proposed techniques to enhance the solution. We introduce the idea of “smart clouds”. Smart clouds refer to point cloud discretizations that are aware of the exact CAD geometry, appropriate to solve a defined problem using a point collocation method and that contain information used to improve locally the solution. We introduce a unified node selection algorithm based on a generalization of the visibility criterion. The proposed algorithm leads to a significant reduction of the error for concave problems and does not have any drawback for convex problems. The point collocation methods rely on many parameters. We select in this thesis parameters for the Generalized Finite Difference (GFD) method and the Discretization-Corrected Particle Strength Exchange (DC PSE) method that we deem appropriate for most problems from the field of linear elasticity. We also show that solution improvement techniques, based on the use of Voronoi diagrams or on a stabilization of the PDE, do not lead to a reduction of the error for all of the considered benchmark problems. These methods shall therefore be used with care. We propose two types of

Computer Aided Design (CAD) software packages are used in the industry to design mechanical systems. Then, calculations are often performed using simulation software packages to improve the quality of the design. To speed up the development costs, companies and research centers have been trying to ease the integration of the computation phase in the design phase. The collocation methods have the potential of easing such integration thanks to their meshless nature. The geometry discretization step which is a key element of all computational method is simplified compared to mesh-based methods such as the finite element method. We propose in this thesis a unified workflow that allows the solution of engineering problems defined by partial differential equations (PDEs) directly from input CAD files. The scheme is based on point collocation methods and proposed techniques to enhance the solution. We introduce the idea of “smart clouds”. Smart clouds refer to point cloud discretizations that are aware of the exact CAD geometry, appropriate to solve a defined problem using a point collocation method and that contain information used to improve locally the solution. We introduce a unified node selection algorithm based on a generalization of the visibility criterion. The proposed algorithm leads to a significant reduction of the error for concave problems and does not have any drawback for convex problems. The point collocation methods rely on many parameters. We select in this thesis parameters for the Generalized Finite Difference (GFD) method and the Discretization-Corrected Particle Strength Exchange (DC PSE) method that we deem appropriate for most problems from the field of linear elasticity. We also show that solution improvement techniques, based on the use of Voronoi diagrams or on a stabilization of the PDE, do not lead to a reduction of the error for all of the considered benchmark problems. These methods shall therefore be used with care. We propose two types of

Computer Aided Design (CAD) software packages are used in the industry to design mechanical systems. Then, calculations are often performed using simulation software packages to improve the quality of the design. To speed up the development costs, companies and research centers have been trying to ease the integration of the computation phase in the design phase. The collocation methods have the potential of easing such integration thanks to their meshless nature. The geometry discretization step which is a key element of all computational method is simplified compared to mesh-based methods such as the finite element method. We propose in this thesis a unified workflow that allows the solution of engineering problems defined by partial differential equations (PDEs) directly from input CAD files. The scheme is based on point collocation methods and proposed techniques to enhance the solution. We introduce the idea of “smart clouds”. Smart clouds refer to point cloud discretizations that are aware of the exact CAD geometry, appropriate to solve a defined problem using a point collocation method and that contain information used to improve locally the solution. We introduce a unified node selection algorithm based on a generalization of the visibility criterion. The proposed algorithm leads to a significant reduction of the error for concave problems and does not have any drawback for convex problems. The point collocation methods rely on many parameters. We select in this thesis parameters for the Generalized Finite Difference (GFD) method and the Discretization-Corrected Particle Strength Exchange (DC PSE) method that we deem appropriate for most problems from the field of linear elasticity. We also show that solution improvement techniques, based on the use of Voronoi diagrams or on a stabilization of the PDE, do not lead to a reduction of the error for all of the considered benchmark problems. These methods shall therefore be used with care. We propose two types of

Finite-volume strategies in fluid-structure interaction problems would be of crucialvimportance in many engineering applications such as in the analysis of reed valves in reciprocating compressors. The efficient implementation of this strategy passes from the formulation of reliable high-order schemes on 3D unstructured meshes. The development of high-order models is essential in bending-dominant problems, where the phenomenon of shear blocking appears. In order to solve this problem, it is possible to either increase the number of elements or increase the interpolation order of the main variable. Increasing the number of elements does not always yield good results and implies a very high computational cost that, in real problems, is inadmissible. Using unstructured meshes is also vital because they are necessary for real problems where the geometries are complex and depart from canonical rectangular or regular shapes. This work presents a series of tests to demonstrate the feasibility of a high-order model using finite volumes for linear elasticity on unstructured and structured meshes. The high-order interpolation will be performed using two different schemes such as the Moving Least Squares (MLS) and the Local Regression Estimators (LRE). The reliability of the method for solving 2D and 3D problems will be verified by solving some known test cases with an analytical solution such as a thin beam or problems where stress concentrations appear., P. Castrillo gratefully acknowledges the Universitat Politècnica de Catalunya and Banco Santander for the financial support of his predoctoral grant FPI-UPC (109 FPI-UPC 2018). The authors are supported by the Ministerio de Economía y Competitividad, Spain, RETOtwin project (PDC2021-120970-I00)., Peer Reviewed, Postprint (published version)

We present an analytic formulation for an elastic medium under uniform stress and in plane strain, with a very large number of cracks that are allowed to intersect. The singularities at the tips of the cracks are represented exactly; the solution is written in terms of series expansions of negative powers of a complex variable defined outside the unit circle in a complex reference plane. We modeled an assembly of 10,000 cracks and show stress trajectories both for the whole model and for individual cracks, chosen at random from the assembly.

This paper presents a high-order finite volume method for solving linear elasticity problems on two-dimensional unstructured meshes. The method is designed to increase the effectiveness of finite volume methods in solving structural problems affected by shear locking. The particular feature of the proposed method is the use of Moving Least Squares (MLS) and Local Regression Estimators (LRE). Unlike other approaches proposed before, these interpolation schemes lead to a natural and simple extension of the classical finite volume method to arbitrary order. The unknowns of the problem are still the nodal values of the displacement which are obtained implicitly in a direct solution strategy. Some canonical tests are performed to demonstrate the accuracy of the method. An analytical example is considered to evaluate the sensitivity of the solution concerning the parameters of the algorithm. A thin curved beam and a crack problem are considered to show that the method can deal with the shear locking effect, stress concentrations, and geometries where unstructured meshes are required. An overall better behavior of the LRE is observed. A comparison between low and high-order schemes is presented, and a set of parameters for the interpolation method is found, delivering good results for the proposed cases., P. Castrillo gratefully acknowledges the Universitat Politècnica de Catalunya and Banco Santander for the financial support of his predoctoral grant FPI-UPC (109 FPI-UPC 2018). A. Canelas thanks the Uruguayan research councils ANII and CSIC for the financial support., Peer Reviewed, Postprint (author's final draft)

Computing the total energy of a system of N interacting dislocations in the presence of arbitrary free surfaces is a difficult task, requiring Finite Element (FE) numerical calculations. Worst, high accuracy requires very fine meshes in the proximity of each dislocation core. Here we show that FE calculations can be conveniently replaced by a Machine Learning (ML) approach. After formulating the elastic problem in terms of one and two-body terms only, we use Sobolev training to obtain consistent information on both energy and forces, fitted using a feed-forward neural network (NN) architecture. As an example, we apply the proposed methodology to corrugated, heteroepitaxial semiconductor films, searching for the minimum-energy dislocation distributions by using Monte Carlo. Importantly, the presence of an interaction cutoff allows for the application of the method to systems of different sizes without the need to repeat training. Millions of energy evaluations are performed, a task which would have been impossible by brute-force FE calculations. Finally, we show how forces can be exploited in running 2D ML-based dislocation dynamics simulations.

We present an analytic formulation for an elastic medium under uniform stress and in plane strain, with a very large number of cracks that are allowed to intersect. The singularities at the tips of the cracks are represented exactly; the solution is written in terms of series expansions of negative powers of a complex variable defined outside the unit circle in a complex reference plane. We modeled an assembly of 10,000 cracks and show stress trajectories both for the whole model and for individual cracks, chosen at random from the assembly.

The article covers the problem of dynamic theory of linear elasticity when seismic wave incidences perpendicular to the axis of long pipe laid in a high embankment and filled with ideal compressible fluid. The design scheme is shown in Fig. 1. Equation of motion in vector form for isotropic body known from the dynamic theory of elasticity has been obtained.

The article covers the problem of dynamic theory of linear elasticity when seismic wave incidences perpendicular to the axis of long pipe laid in a high embankment and filled with ideal compressible fluid. The design scheme is shown in Fig. 1. Equation of motion in vector form for isotropic body known from the dynamic theory of elasticity has been obtained.

The article covers the problem of dynamic theory of linear elasticity when seismic wave incidences perpendicular to the axis of long pipe laid in a high embankment and filled with ideal compressible fluid. The design scheme is shown in Fig. 1. Equation of motion in vector form for isotropic body known from the dynamic theory of elasticity has been obtained.

The article covers the problem of dynamic theory of linear elasticity when seismic wave incidences perpendicular to the axis of long pipe laid in a high embankment and filled with ideal compressible fluid. The design scheme is shown in Fig. 1. Equation of motion in vector form for isotropic body known from the dynamic theory of elasticity has been obtained.

The article covers the problem of dynamic theory of linear elasticity when seismic wave incidences perpendicular to the axis of long pipe laid in a high embankment and filled with ideal compressible fluid. The design scheme is shown in Fig. 1. Equation of motion in vector form for isotropic body known from the dynamic theory of elasticity has been obtained.

The article covers the problem of dynamic theory of linear elasticity when seismic wave incidences perpendicular to the axis of long pipe laid in a high embankment and filled with ideal compressible fluid. The design scheme is shown in Fig. 1. Equation of motion in vector form for isotropic body known from the dynamic theory of elasticity has been obtained.

The article covers the problem of dynamic theory of linear elasticity when seismic wave incidences perpendicular to the axis of long pipe laid in a high embankment and filled with ideal compressible fluid. The design scheme is shown in Fig. 1. Equation of motion in vector form for isotropic body known from the dynamic theory of elasticity has been obtained.

The article covers the problem of dynamic theory of linear elasticity when seismic wave incidences perpendicular to the axis of long pipe laid in a high embankment and filled with ideal compressible fluid. The design scheme is shown in Fig. 1. Equation of motion in vector form for isotropic body known from the dynamic theory of elasticity has been obtained.

Composite structures, such as laminated beams, plates and shells, are widely used in the automotive, aerospace and marine industries due to their superior specific strength and tailor-able mechanical properties. Because of their use in a wide range of applications, and their commonplace in the engineering design community, the need to accurately predict their behavior to external stimuli is crucial. We consider in this thesis the application of the least-squares finite element method (LSFEM) to problems of static deformations of laminated and sandwich plates and transient plane stress deformations of sandwich beams. Models are derived to express the governing equations of linear elasticity in terms of layer-wise continuous variables for composite plates and beams, which allow inter-laminar continuity conditions at layer interfaces to be satisfied. When Legendre-Gauss-Lobatto (LGL) basis functions with the LGL nodes taken as integration points are used to approximate the unknown field variables, the methodology yields a system of discrete equations with a symmetric positive definite coefficient matrix. The main goal of this research is to determine the efficacy of the LSFEM in accurately predicting stresses in laminated composites when subjected to both quasi-static and transient surface tractions. Convergence of the numerical algorithms with respect to the LGL basis functions in space and time (when applicable) is also considered and explored. In the transient analysis of sandwich beams, we study the sensitivity of the first failure load to the beam's aspect ratio (AR), facesheet-core thickness ratio (FCTR) and facesheet-core stiffness ratio (FCSR). We then explore how failure of sandwich beams is affected by considering facesheet and core materials with different in-plane and transverse stiffness ratios. Computed results are compared to available analytical solutions, published results and those found by using the commercial FE software ABAQUS where appropriate

Energy transition extends the range of geological settings and physical processes to be taken into account in subsurface reservoir modelling. Many of these applications consider essentially anisotropic reservoir or require advanced gridding that can not be resolved consistently by conventionally used Two Point Flux Approximation (TPFA). In this project we present a Nonlinear Two Point Flux Approximation (NTPFA) based on gradient reconstruction and homogenization function. The approximation provides consistent solution for full permeability tensor on various grids. The approach combines flux guesses in a nonlinear way such that the obtained approximation is essentially monotone that guarantees the positivity of solution. We demonstrate the consistency of approach on several examples. We also use the multi-physics capabilities to test the simulator on saturation transport of dead oil when displaced with water. The developed approximation was implemented within Delft Advanced Research Terra Simulator (DARTS). Next we propose a new Nonlinear Two Point Stress Approximation technique which follows the collocated finite volume scheme for mechanical problem. In this section we try to discretize the linear elasticity equation by using nonlinear traction flux at interfaces similar to the setup used in fluid flow problem. This is done by balancing each component of traction individually and using the weighting scheme suggested in flux approximations., Civil Engineering

Das octAFEM3D Softwarepaket dient der numerischen Lösung von partiellen Differentialgleichungen. Drei lineare Modellprobleme in drei Raumdimensionen können mit einer Least-Squares Finiten-Elemente-Methode niedrigsten Polynomgrades gelöst werden. Die Approximation von inhomogenen Randdaten ist möglich. Die adaptive Netzverfeinerung wird mit einer kollektiven Markierungsstrategie umgesetzt. Die Implementierung basiert auf dem AFEM Softwarepaket der Arbeitsgruppe der Numerischen Mathematik von Prof. Carsten Carstensen an der Humboldt-Universität zu Berlin. Die Programme wurden implementiert und getestet für die Matlab Version 9.6.0.1072779 (R2019a) und Octave Version 5.1.0. Dieses Softwarepaket ist Teil der Dissertation „Adaptive least-squares finite element method with optimal convergence rates“ von Philipp Bringmann an der Humboldt-Universität zu Berlin unter der Betreuung von Prof. Carsten Carstensen., This is the octAFEM3D package for the numerical solution of partial differential equations. Three linear model problems in three spatial dimensions can be solved by a lowest-order least-squares finite element method. The approximation of inhomogeneous boundary conditions is included. Adaptive mesh-refinement is realised with a collective marking strategy. The software is derived from the AFEM package of the numerical analysis working group of Prof. Carsten Carstensen at Humboldt-Universität zu Berlin. The code is implemented and tested for Matlab 9.6.0.1072779 (R2019a) and Octave 5.1.0. This software package is part of the PhD thesis „Adaptive least-squares finite element method with optimal convergence rates“ by Philipp Bringmann at Humboldt-Universität zu Berlin under the supervision of Prof. Carsten Carstensen.

In this paper, the study of linear elasticity is proposed in a nano reinforced composite material subjected to a constant external force in order to be used in the manufacture of lower limb prostheses, for which a computational algorithm was developed that solves the equation of linear elasticity (Navier equation), using the boundary element method and radial base functions. It was determined if the use of an algorithm can predict the change in a two-dimensional geometry at the level of deformations, displacements and stresses in a composite material reinforced with carbon nanotubes used in the manufacture of lower limb prostheses and demonstrate compliance with the desired requirements when subjected to constant force. Therefore, with the implementation of the algorithm and the analysis of the information obtained, the selection process of a nano-reinforced composite material for use in the development of lower limb prostheses is supported when subjected to constant force. According to the algorithm developed and the results found, the boundary element method allows the simulation of the mechanical behavior of a composite material (A36 steel, at carbon nanotube concentrations of 1%, 2%,3%). The stress and deformation calculation in the model was performed using an algorithm that calculates the derivatives using the multi-quadratic function of the radial-basis functions. With this study, when numerically solving the catile verbeam problem by varying the concentrations of the material, the stress-deformation characteristic graphs were generated, which allowed determining that with the chosen concentrations a good mechanical response occurs that allows considering the material to be used in the development of lower limb prostheses., En este paper se plantea el estudio de elasticidad lineal en un material compuesto nano reforzado sometido a una fuerza externa constante con el fin de ser utilizado en la fabricación de prótesis de miembro inferior, para lo cual se desarrolló un algoritmo computacional que resuelve la ecuación de elasticidad lineal (ecuación de Navier), utilizando el método de elementos de frontera y funciones de base radial. Se determinó si el uso de un algoritmo puede predecir el cambio en una geometría bidimensional a nivel de deformaciones, desplazamientos y esfuerzos en un material compuesto reforzado con nanotubos de carbono; utilizado en la fabricación de prótesis de miembro inferior y evidenciar el cumplimiento de los requerimientos deseados al ser sometido a una fuerza constante. Por lo anterior el análisis de la información obtenida, se apoya el proceso de selección de un material compuesto nanoreforzado para uso en el desarrollo de prótesis de miembro inferior al ser sometido a una fuerza constante. Según el algoritmo desarrollado y los resultados encontrados, el método de elementos de frontera permite la simulación del comportamiento mecánico de un material compuesto (acero A36, a concentraciones de nanotubos de carbono de 1%, 2%, 3%).

The article covers the problem of dynamic theory of linear elasticity when seismic wave incidences perpendicular to the axis of long pipe laid in a high embankment and filled with ideal compressible fluid. The design scheme is shown in Fig. 1. Equation of motion in vector form for isotropic body known from the dynamic theory of elasticity has been obtained.

In this paper, the study of linear elasticity is proposed in a nano reinforced composite material subjected to a constant external force in order to be used in the manufacture of lower limb prostheses, for which a computational algorithm was developed that solves the equation of linear elasticity (Navier equation), using the boundary element method and radial base functions. It was determined if the use of an algorithm can predict the change in a two-dimensional geometry at the level of deformations, displacements and stresses in a composite material reinforced with carbon nanotubes used in the manufacture of lower limb prostheses and demonstrate compliance with the desired requirements when subjected to constant force. Therefore, with the implementation of the algorithm and the analysis of the information obtained, the selection process of a nano-reinforced composite material for use in the development of lower limb prostheses is supported when subjected to constant force. According to the algorithm developed and the results found, the boundary element method allows the simulation of the mechanical behavior of a composite material (A36 steel, at carbon nanotube concentrations of 1%, 2%,3%). The stress and deformation calculation in the model was performed using an algorithm that calculates the derivatives using the multi-quadratic function of the radial-basis functions. With this study, when numerically solving the catile verbeam problem by varying the concentrations of the material, the stress-deformation characteristic graphs were generated, which allowed determining that with the chosen concentrations a good mechanical response occurs that allows considering the material to be used in the development of lower limb prostheses., En este paper se plantea el estudio de elasticidad lineal en un material compuesto nano reforzado sometido a una fuerza externa constante con el fin de ser utilizado en la fabricación de prótesis de miembro inferior, para lo cual se desarrolló un algoritmo computacional que resuelve la ecuación de elasticidad lineal (ecuación de Navier), utilizando el método de elementos de frontera y funciones de base radial. Se determinó si el uso de un algoritmo puede predecir el cambio en una geometría bidimensional a nivel de deformaciones, desplazamientos y esfuerzos en un material compuesto reforzado con nanotubos de carbono; utilizado en la fabricación de prótesis de miembro inferior y evidenciar el cumplimiento de los requerimientos deseados al ser sometido a una fuerza constante. Por lo anterior el análisis de la información obtenida, se apoya el proceso de selección de un material compuesto nanoreforzado para uso en el desarrollo de prótesis de miembro inferior al ser sometido a una fuerza constante. Según el algoritmo desarrollado y los resultados encontrados, el método de elementos de frontera permite la simulación del comportamiento mecánico de un material compuesto (acero A36, a concentraciones de nanotubos de carbono de 1%, 2%, 3%).

Thin-tile vaults are a a type of vaults that went out of fashion in the early twentieth century. Its origins are around the Mediterranean, but modern interest is mostly due to Guastavino, and the research done at MIT and ETH. The thin-tile vaults have a unique construction method without any temporary support. Eventually the increase in labour costs and the advancements in concrete and steel made the structure non-competitive.Robotics are a type of machines that can perform (semi)-automated tasks. In the past decades the development of robots have led to their implementation in the construction industry. Robots developed specifically for masonry show a high promise where they're able to lay much more bricks than even the most skilled mason.This research aims to investigate the time it takes for a robot to construct a thin-tile vault, and thereafter to advance the possibilities of research into and construction of the thin-tile vault. This is researched by answering the following question:How does a robotic construction of a parametrically designed thin-tile vault perform based on step-wise structural analyses?To answer this question a parametric model has been made to include the design of the thin-tile vault, the structural analysis and the robotic construction. This model is made in Grasshopper, but is supported by Python and RoboDK. Python performs any calculation necessary and analysis of this output. RoboDK is a robotic simulation program aimed to give users a tool to translate their design to robotic instructions. The structure of this report is based on the three aspects of the parametric model.In the first part the state of the art and the relevance of this research is stated. Thereafter the objectives, questions and methodology are noted.The first part of the model is related to the design. In the first chapter a literature review is provided on thin-tile vaults and similar vault structures. The second chapter of this part describes how the design model has, Civil Engineering | Building Engineering

We consider mechanics of composite materials in which thin inclusions are modeled by lower-dimensional manifolds. By successively applying the dimensional reduction to junctions and intersections within the material, a geometry of hierarchically connected manifolds is formed which we refer to as mixed-dimensional. The governing equations with respect to linear elasticity are then defined on this mixed-dimensional geometry. The resulting system of partial differential equations is also referred to as mixed-dimensional, since functions defined on domains of multiple dimensionalities are considered in a fully coupled manner. With the use of a semi-discrete differential operator, we obtain the variational formulation of this system in terms of both displacements and stresses. The system is then analyzed and shown to be well-posed with respect to appropriately weighted norms. Numerical discretization schemes are proposed using well-known mixed finite elements in all dimensions. The schemes conserve linear momentum locally while relaxing the symmetry condition on the stress tensor. Stability and convergence are shown using a priori error estimates and confirmed numerically., QC 20210318

The derivation of nonlocal strong forms for many physical problems remains cumbersome in traditional methods. In this paper, we apply the variational principle/weighted residual method based on nonlocal operator method for the derivation of nonlocal forms for elasticity, thin plate, gradient elasticity, electro-magneto-elasticity and phase-field fracture method. The nonlocal governing equations are expressed as an integral form on support and dual-support. The first example shows that the nonlocal elasticity has the same form as dual-horizon non-ordinary state-based peridynamics. The derivation is simple and general and it can convert efficiently many local physical models into their corresponding nonlocal forms. In addition, a criterion based on the instability of the nonlocal gradient is proposed for the fracture modelling in linear elasticity. Several numerical examples are presented to validate nonlocal elasticity and the nonlocal thin plate. © 2021, The Author(s).

Composite structures, such as laminated beams, plates and shells, are widely used in the automotive, aerospace and marine industries due to their superior specific strength and tailor-able mechanical properties. Because of their use in a wide range of applications, and their commonplace in the engineering design community, the need to accurately predict their behavior to external stimuli is crucial. We consider in this thesis the application of the least-squares finite element method (LSFEM) to problems of static deformations of laminated and sandwich plates and transient plane stress deformations of sandwich beams. Models are derived to express the governing equations of linear elasticity in terms of layer-wise continuous variables for composite plates and beams, which allow inter-laminar continuity conditions at layer interfaces to be satisfied. When Legendre-Gauss-Lobatto (LGL) basis functions with the LGL nodes taken as integration points are used to approximate the unknown field variables, the methodology yields a system of discrete equations with a symmetric positive definite coefficient matrix. The main goal of this research is to determine the efficacy of the LSFEM in accurately predicting stresses in laminated composites when subjected to both quasi-static and transient surface tractions. Convergence of the numerical algorithms with respect to the LGL basis functions in space and time (when applicable) is also considered and explored. In the transient analysis of sandwich beams, we study the sensitivity of the first failure load to the beam's aspect ratio (AR), facesheet-core thickness ratio (FCTR) and facesheet-core stiffness ratio (FCSR). We then explore how failure of sandwich beams is affected by considering facesheet and core materials with different in-plane and transverse stiffness ratios. Computed results are compared to available analytical solutions, published results and those found by using the commercial FE software ABAQUS where appropriate

This work was supported by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 ‚Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis’ under the project ‚Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics‘. The files triangulation/Node.m and triangulation/Simplex.m are Matlab implementations of the corresponding classes from C. T. Traxler „An algorithm for adaptive mesh refinement in n dimensions.“ Computing, 59(2):115-137, 1997. The remaining files are provided under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 or (at your option) any later version. See LICENSE.md for further details., Das octAFEM3D Softwarepaket dient der numerischen Lösung von partiellen Differentialgleichungen. Drei lineare Modellprobleme in drei Raumdimensionen können mit einer Least-Squares Finiten-Elemente-Methode niedrigsten Polynomgrades gelöst werden. Die Approximation von inhomogenen Randdaten ist möglich. Die adaptive Netzverfeinerung wird mit einer kollektiven Markierungsstrategie umgesetzt. Die Implementierung basiert auf dem AFEM Softwarepaket der Arbeitsgruppe der Numerischen Mathematik von Prof. Carsten Carstensen an der Humboldt-Universität zu Berlin. Die Programme wurden implementiert und getestet für die Matlab Version 9.6.0.1072779 (R2019a) und Octave Version 5.1.0. Dieses Softwarepaket ist Teil der Dissertation „Adaptive least-squares finite element method with optimal convergence rates“ von Philipp Bringmann an der Humboldt-Universität zu Berlin unter der Betreuung von Prof. Carsten Carstensen., This is the octAFEM3D package for the numerical solution of partial differential equations. Three linear model problems in three spatial dimensions can be solved by a lowest-order least-squares finite element method. The approximation of inhomogeneous boundary conditions is included. Adaptive mesh-refinement is realised with a collective marking strategy. The software is derived from the AFEM package of the numerical analysis working group of Prof. Carsten Carstensen at Humboldt-Universität zu Berlin. The code is implemented and tested for Matlab 9.6.0.1072779 (R2019a) and Octave 5.1.0. This software package is part of the PhD thesis „Adaptive least-squares finite element method with optimal convergence rates“ by Philipp Bringmann at Humboldt-Universität zu Berlin under the supervision of Prof. Carsten Carstensen.

Metoda konečných prvků je bezpochyby jedna z nejrozšířenějších metod pro řešení úloh mechaniky pevných těles. Nicméně, jedná se o metodu aproximační a její výsledky jsou závislé na definici prvku použitého pro výpočet. Nejjednodušší prvky s jedním primárním polem často trpí takzvaným „zamykáním“, tedy přílišnou tuhostí při ohybovém namáhání nebo pokud je těleso tvořeno nestlačitelným materiálem. V takovém případě je alternativou použití prvku o více neznámých polích. Článek představuje jeden z prvků o dvou neznámých polích formulovaný na základě Hellinger-Reissner variačního principu a na příkladech porovnává jeho robustnost s ostatními metodami, které byly v minulosti použity pro odstranění zamykání. Úlohy jsou řešeny v rámci lineární elasticity., The Finite Element Method is without a doubt one of the most prominent tools in solving the equations governing mechanics of solids. It is an approximative method and, as such, its performance largely depends on the definition of the finite element used in a computation. The simplest elements, based on one primary field, tend to suffer from “locking”, that is excessive stiffness when an element is subjected to bending or the material is nearing the limit of incompressibility. One of the alternatives is the use of an element based on multiple primary fields. The present article aims to describe one such element (based on mixed-field Hellinger-Reissner variational principle) and analyze its robustness in comparison to other methods which were used in the past to mitigate locking. The analysis will be done in the framework of linear elastostatics.

Metoda konečných prvků je bezpochyby jedna z nejrozšířenějších metod pro řešení úloh mechaniky pevných těles. Nicméně, jedná se o metodu aproximační a její výsledky jsou závislé na definici prvku použitého pro výpočet. Nejjednodušší prvky s jedním primárním polem často trpí takzvaným „zamykáním“, tedy přílišnou tuhostí při ohybovém namáhání nebo pokud je těleso tvořeno nestlačitelným materiálem. V takovém případě je alternativou použití prvku o více neznámých polích. Článek představuje jeden z prvků o dvou neznámých polích formulovaný na základě Hellinger-Reissner variačního principu a na příkladech porovnává jeho robustnost s ostatními metodami, které byly v minulosti použity pro odstranění zamykání. Úlohy jsou řešeny v rámci lineární elasticity., The Finite Element Method is without a doubt one of the most prominent tools in solving the equations governing mechanics of solids. It is an approximative method and, as such, its performance largely depends on the definition of the finite element used in a computation. The simplest elements, based on one primary field, tend to suffer from “locking”, that is excessive stiffness when an element is subjected to bending or the material is nearing the limit of incompressibility. One of the alternatives is the use of an element based on multiple primary fields. The present article aims to describe one such element (based on mixed-field Hellinger-Reissner variational principle) and analyze its robustness in comparison to other methods which were used in the past to mitigate locking. The analysis will be done in the framework of linear elastostatics.

This work was supported by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 ‚Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis’ under the project ‚Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics‘., Die Least-Squares Finite-Elemente-Methoden (LSFEMn) basieren auf der Minimierung des Least-Squares-Funktionals, das aus quadrierten Normen der Residuen eines Systems von partiellen Differentialgleichungen erster Ordnung besteht. Dieses Funktional liefert einen a posteriori Fehlerschätzer und ermöglicht die adaptive Verfeinerung des zugrundeliegenden Netzes. Aus zwei Gründen versagen die gängigen Methoden zum Beweis optimaler Konvergenzraten, wie sie in Carstensen, Feischl, Page und Praetorius (Comp. Math. Appl., 67(6), 2014) zusammengefasst werden. Erstens scheinen fehlende Vorfaktoren proportional zur Netzweite den Beweis einer schrittweisen Reduktion der Least-Squares-Schätzerterme zu verhindern. Zweitens kontrolliert das Least-Squares-Funktional den Fehler der Fluss- beziehungsweise Spannungsvariablen in der H(div)-Norm, wodurch ein Datenapproximationsfehler der rechten Seite f auftritt. Diese Schwierigkeiten führten zu einem zweifachen Paradigmenwechsel in der Konvergenzanalyse adaptiver LSFEMn in Carstensen und Park (SIAM J. Numer. Anal., 53(1), 2015) für das 2D-Poisson-Modellproblem mit Diskretisierung niedrigster Ordnung und homogenen Dirichlet-Randdaten. Ein neuartiger expliziter residuenbasierter Fehlerschätzer ermöglicht den Beweis der Reduktionseigenschaft. Durch separiertes Markieren im adaptiven Algorithmus wird zudem der Datenapproximationsfehler reduziert. Die vorliegende Arbeit verallgemeinert diese Techniken auf die drei linearen Modellprobleme das Poisson-Problem, die Stokes-Gleichungen und das lineare Elastizitätsproblem. Die Axiome der Adaptivität mit separiertem Markieren nach Carstensen und Rabus (SIAM J. Numer. Anal., 55(6), 2017) werden in drei Raumdimensionen nachgewiesen. Die Analysis umfasst Diskretisierungen mit beliebigem Polynomgrad sowie inhomogene Dirichlet- und Neumann-Randbedingungen. Abschließend bestätigen numerische Experimente mit dem h-adaptiven Algorithmus die theoretisch bewiesenen optimalen Konvergenzraten., The least-squares finite element methods (LSFEMs) base on the minimisation of the least-squares functional consisting of the squared norms of the residuals of first-order systems of partial differential equations. This functional provides a reliable and efficient built-in a posteriori error estimator and allows for adaptive mesh-refinement. The established convergence analysis with rates for adaptive algorithms, as summarised in the axiomatic framework by Carstensen, Feischl, Page, and Praetorius (Comp. Math. Appl., 67(6), 2014), fails for two reasons. First, the least-squares estimator lacks prefactors in terms of the mesh-size, what seemingly prevents a reduction under mesh-refinement. Second, the first-order divergence LSFEMs measure the flux or stress errors in the H(div) norm and, thus, involve a data resolution error of the right-hand side f. These difficulties led to a twofold paradigm shift in the convergence analysis with rates for adaptive LSFEMs in Carstensen and Park (SIAM J. Numer. Anal., 53(1), 2015) for the lowest-order discretisation of the 2D Poisson model problem with homogeneous Dirichlet boundary conditions. Accordingly, some novel explicit residual-based a posteriori error estimator accomplishes the reduction property. Furthermore, a separate marking strategy in the adaptive algorithm ensures the sufficient data resolution. This thesis presents the generalisation of these techniques to three linear model problems, namely, the Poisson problem, the Stokes equations, and the linear elasticity problem. It verifies the axioms of adaptivity with separate marking by Carstensen and Rabus (SIAM J. Numer. Anal., 55(6), 2017) in three spatial dimensions. The analysis covers discretisations with arbitrary polynomial degree and inhomogeneous Dirichlet and Neumann boundary conditions. Numerical experiments confirm the theoretically proven optimal convergence rates of the h-adaptive algorithm.

Energy transition extends the range of geological settings and physical processes to be taken into account in subsurface reservoir modelling. Many of these applications consider essentially anisotropic reservoir or require advanced gridding that can not be resolved consistently by conventionally used Two Point Flux Approximation (TPFA). In this project we present a Nonlinear Two Point Flux Approximation (NTPFA) based on gradient reconstruction and homogenization function. The approximation provides consistent solution for full permeability tensor on various grids. The approach combines flux guesses in a nonlinear way such that the obtained approximation is essentially monotone that guarantees the positivity of solution. We demonstrate the consistency of approach on several examples. We also use the multi-physics capabilities to test the simulator on saturation transport of dead oil when displaced with water. The developed approximation was implemented within Delft Advanced Research Terra Simulator (DARTS). Next we propose a new Nonlinear Two Point Stress Approximation technique which follows the collocated finite volume scheme for mechanical problem. In this section we try to discretize the linear elasticity equation by using nonlinear traction flux at interfaces similar to the setup used in fluid flow problem. This is done by balancing each component of traction individually and using the weighting scheme suggested in flux approximations., Civil Engineering

The aim of this work is to present an overview about the combination of the Reduced Basis Method (RBM) with two different approaches for Fluid–Structure Interaction (FSI) problems, namely a monolithic and a partitioned approach. We provide the details of implementation of two reduction procedures, and we then apply them to the same test case of interest. We first implement a reduction technique that is based on a monolithic procedure where we solve the fluid and the solid problems all at once. We then present another reduction technique that is based on a partitioned (or segregated) procedure: the fluid and the solid problems are solved separately and then coupled using a fixed point strategy. The toy problem that we consider is based on the Turek–Hron benchmark test case, with a fluid Reynolds number Re=100.