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In the study of some real cases, it is possible to encounter well-defined geometric conditions, of an industrial or design type—for example, the case of a specific volume within each of several holes. In most of these cases, it is recommended to fulfil a function defined in a domain in which information is missing in one or more sub-domains (holes) of the global set, where the function data are not known. The problem of filling holes or completing a surface in three dimensions appears in many fields of computing, such as computer-aided geometric design (CAGD). A method to solve the shape-preserving variational spline approximation problem for hole filling in generalized offset surfaces is presented. The existence and uniqueness of the solution of the studied method are established, as well as the computation, and certain convergence results are analyzed. A graphic and numerical example complete this study to demonstrate the effectiveness of the presented method. This manuscript presents the resolution of a complicated problem due to the study of some criteria that can be traduced via an approximation problem related to generalized offset surfaces with holes and also the preservation of the shape of such surfaces. [ABSTRACT FROM AUTHOR]

Standard Offset surfaces are defined as locus of the points which are at constant distance along the unit normal direction from the generator surfaces. Offset are widely used in various practical applications, such as tolerance analysis, geometric optics and robot path-planning. In some of the engineering applications, we need to extend the concept of standard offset to the generalized offset where distance offset is not necessarily constant and offset direction are not necessarily along the normal direction. Normally, a generalized offset is functionally more complex than its progenitor because of the square root appears in the expression of the unit normal vector. For this, an approximation method of its construction is necessary. In many situation it is necessary to fill or reconstruct certain function defined in a domain in which there is a lack of information inside one or several sub-domains (holes). In some practical cases, we may have some specific geometrical constrains, of industrial or design type, for example, the case of a specified volume inside each one of these holes. The problem of filling holes or completing a 3D surface arises in all sorts of computational graphics areas, like CAGD, CAD-CAM, Earth Sciences, computer vision in robotics, image reconstruction from satellite and radar information, etc. In this work we present an approximation method of filling holes of the generalized offset of a surface when there is a lack information in a sub-domain of the function that define it. We prove the existence and uniqueness of solution of this problem, we show how to compute it and we establish a convergence result of this approximation method. Finally, we give some graphical and numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we propose a variational method in order to solve Bratu's problem for two dimensions in an adequate space of biquadratic spline functions. The solution is obtained by resolving a sequence of boundary value problems. We study some characterizations of the functions of such sequence and we express them as some linear combination of biquadratic spline bases functions. We finish by showing some numerical and graphical examples in order to prove the validity and the effectiveness of our method. [ABSTRACT FROM AUTHOR]

We present an approximation problem of surfaces of a generalized offset surface with offset variable distances and directions. Such approximating surface fits some given data points and minimizes a Sobolev's semi-norm of order 3. The study of the new results, from a mathematical point of view, carefully establishing the proof of the convergence between the generalized offset surface and its approximating spline in an adequate parametric bicubic spline space. Moreover, the approximating spline function is computed and an estimation of the relative error is introduced. Finally, some numerical and graphic examples are shown in order to prove the useful and the effectiveness of our method. [ABSTRACT FROM AUTHOR]

Abstract We present a smoothing method of a set of points to be approximated from a given boundary value problem for the modified Helmholtz equation. A solution of such problem will be called MH-spline. Two convergence results are carefully established. Some numerical and graphic examples are shown in order to test the validity of the method. [ABSTRACT FROM AUTHOR]

This article deals with a numerical approximation method using an evolutionary partial differential equation (PDE) by discrete variational splines in a finite element space. To formulate the problem, we need an evolutionary PDE equation with respect to the time and the position, certain boundary conditions and a set of approximating points. We show the existence and uniqueness of the solution and we study a computational method to compute such a solution. Moreover, we established a convergence result with respect to the time and the position. We provided several numerical and graphic examples of approximation in order to show the validity and effectiveness of the presented method.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 5-18, 2018 [ABSTRACT FROM AUTHOR]

In this paper we present a variational approximation method for solving Troesch's problem. The existence and the uniqueness of this problem are shown. Moreover, we construct a sequence of approximate solutions of the problem from the number of knots of a partition of the domain. Such sequence converges to the exact solution of the problem. Finally, we analyze some graphical and numerical examples in order to show the efficiency of our method. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with an approximation problem concerning vector fields through the new notion of div–rot variational splines. The minimizing problem is addressed in a finite element space through the choice of some semi-norms based on decomposition of the divergence operator and vector fields into a form with a rotational part. We study the existence and the uniqueness of the solution of such a problem. Then, a convergence result and an estimation of the error are established. Some numerical and graphical examples are analyzed in order to prove the validity of our method. Furthermore, we compare and show how our method improves upon one existing in the literature. [Copyright &y& Elsevier]

Abstract: This paper deals with the construction and characterization of discrete variational PDE splines. To formulate the problem, we need an evolutionary PDE equation, certain boundary conditions and a set of points to approximate. We show the existence and the uniqueness of the solution of such a problem and we establish a convergence result of a discrete variational evolutionary PDE spline to a given function. We present several numerical and graphical examples of construction and approximation of surfaces in order to prove the validity of the presented method. [Copyright &y& Elsevier]

Abstract: In this paper, we study the continuity of surfaces by a shape-blending process. Furthermore, we study the continuity of the ruled surfaces constructed by linear interpolation between two pairs of continuous curves. We give some conditions for the continuity of composite surfaces in a shape-blending process. A practical approach is proposed to maintain the continuity of Bézier surfaces pairs in a shape-blending process by adjusting the control points along the common boundary of the resulting surface-pair. We finish by proving and justifying the efficiency of the approaching method with some graphical examples. [Copyright &y& Elsevier]

Abstract: An approximation problem of parametric curves and surfaces is studied by a new kind of spline functions from some Lagrange or Hermite data set. We present an interpolation problem by minimizing a functional on a parametric finite element space in order to obtain the new notion of a spline. We call it discrete interpolating variational spline. We show how to compute in practice such spline and we carefully prove a convergence result. To illustrate the generality and practice of this work we give some particular cases and we finish by presenting some numerical and graphical examples. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a new approach for solving second kind Fredholm integral equation system. The method is based on the minimization of a suitable functional in a certain Sobolev space. We study the convergence of the method and we present some numerical examples in order to show the validity of the proposed method. [Copyright &y& Elsevier]

Abstract: Smoothing of noisy data has always been a topic of interest in many areas where computer simulations have been performed, including natural sciences as well as economics and social sciences. In this paper we present an approximation method of explicit curves or surfaces from exact and noisy data by fairness cubic or bicubic splines. A variational problem of explicit curves or surfaces is obtained by minimizing a quadratic functional in a space of cubic or bicubic splines from noisy data. We show the existence and uniqueness of this problem as long as a convergence result especially for noisy data is carefully established. We analyze some numerical and graphical examples using fictional noisy data in order to prove the validity of our method. [Copyright &y& Elsevier]

Abstract: In this paper, we present an interpolation method for curves from a data set by means of the optimization of the parameters of a quadratic functional in a space of parametric cubic spline functions. The existence and the uniqueness of this problem are shown. Moreover, a convergence result of the method is established in order to justify the method presented. The aforementioned functional involves some real non-negative parameters; the optimal parametric curve is obtained by the suitable optimization of these parameters. Finally, we analyze some numerical and graphical examples in order to show the efficiency of our method. [Copyright &y& Elsevier]

Abstract: In this paper we present an approximation problem of parametric curves and surfaces from a Lagrange or Hermite data set. In particular, we study an interpolation problem by minimizing some functional on a Sobolev space that produces the new notion of interpolating variational spline. We carefully establish a convergence result. Some specific cases illustrate the generality of this work. [Copyright &y& Elsevier]

Abstract: This paper addresses the problem of constructing some free-form curves and surfaces from given to different types of data: exact and noisy data. We extend the theory of -splines over a bounded domain for noisy data to the smoothing variational vector splines. Both results of convergence for respectively the exact and noisy data are established, as soon as some estimations of errors are given. [Copyright &y& Elsevier]

Abstract: This is a note on the paper [A. Kouibia, A.J. López-Linares, M. Pasadas, Approximation of discontinuous curves and surfaces with tangent conditions, J. Comput. Appl. Math. 193 (2006) 51–64]. We consider the constructing problem of a discontinuous parametric curve or surface from a finite set of points and tangent conditions. We develop a method based on the theory of discrete smoothing variational splines conveniently adapted to introduce the tangent conditions and the discontinuity set. We give a convergence result and we analyze some numerical and graphical examples in order to illustrate the effectiveness of the presented method. [Copyright &y& Elsevier]

Abstract: We deal with a smoothing method of constructing some discontinuous curve or surface from a Lagrangean data and tangent conditions. Such method is based on the theory of smoothing variational splines conveniently adapted to introduce the tangent condition and the discontinuity set. To show the efficiency of our method we finish this work by a convergence result and some numerical and graphical examples. [Copyright &y& Elsevier]

We present in this paper a discrete problem of constructing some parametric surfaces with parallelism conditions from some given Lagrangean data. We consider the problem of fitting some scattered points with a surface pseudo-parallel to a given reference surface in a finite element space. Some convergence results are shown. Finally, we analyze some graphical examples in order to prove the validity and the effectiveness of this method. [Copyright &y& Elsevier]

This paper deals with a shape preserving method of interpolating positive data at points of the plane in R2. We formulate a problem in order to define a positive interpolation variational spline in the Sobolev space Hm(Ω), where Ω is a non-empty bounded set of R2, by minimizing the semi-norm of order m, and we discrete such problem in a finite element space. An algorithm allows us to compute the resulting function. Some convergence theorems are established. The error is of the order O(1/N)m when N tends to +∞, being N the number of the interpolating points. Some numerical and graphical examples are given in order to test the validity of this method. [Copyright &y& Elsevier]

We present in this paper an approximation method of curves from sets of Lagrangian data and vectorial tangent subspaces. We define a discrete smoothing fairness spline with tangent conditions by minimizing certain quadratic functional on finite element spaces. Convergence theorem is established and some numerical and graphical examples are analyzed in order to show the validity and the effectiveness of this paper. [Copyright &y& Elsevier]

In this paper we propose an approximation method for solving second kind Volterra integral equation systems by radial basis functions. It is based on the minimization of a suitable functional in a discrete space generated by compactly supported radial basis functions of Wendland type. We prove two convergence results, and we highlight this because most recent published papers in the literature do not include any. We present some numerical examples in order to show and justify the validity of the proposed method. Our proposed technique gives an acceptable accuracy with small use of the data, resulting also in a low computational cost. [ABSTRACT FROM AUTHOR]