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A discrete Ritz method (DRM) is presented for buckling analysis of plates with arbitrary geometric configurations employing the concept of representing structural energy with global variable stiffness. In the DRM approach, a minimum rectangular domain covering the plate is generated, and openings are created within this domain based on the geometric boundaries of the plate. By applying the Gauss-Legendre quadrature rule and generating sufficient Gaussian points within the rectangular domain, along with assigning zero stiffness and thickness for the Gaussian points in the cutouts, the energy of plates with arbitrary geometric configurations can be numerically simulated. This methodology transforms the rectangular domain into a discrete model of a variable stiffness system, enabling the prediction of the deformation of plates of any shape. Furthermore, a prebuckling in-plane stress field is calculated and integrated into the eigenvalue buckling analysis of plates. The DRM formulation ensures that the energy functionals and computation procedures remain standardized and unaffected by variations in plate geometry, thus overcoming the limitations of traditional Ritz methods, which cannot be applied to complex geometric domains. To validate the efficacy of DRM, five numerical examples of plates with distinct geometries are adopted, and the results are compared with those from the literature and the finite element method. Numerical results show that DRM is capable of handling of plates with complex geometries and distinct boundary conditions, and DRM results are almost consistent with those of existing literature, indicating the feasibility and stability of DRM. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new numerical method to simulate acoustic scattering problems in two‐dimensional periodic structures with non‐periodic incident fields. Applying the Floquet‐Bloch transform to the scattering problem yields a family of quasi‐periodic boundary value problems dependent on the Floquet‐Bloch parameter. Consequently, the solution of the original scattering problem is written as the inverse Floquet‐Bloch transform of the solutions to these boundary value problems. The key step in our method is the numerical approximation of this integral transform by a quadrature rule with a nonuniform choice of quadrature points adapted to the regularity of the family of quasi‐periodic solutions. This achieved by a graded subdivision of the full interval for the Floquet‐Bloch parameter and applying a Gauss‐Legrendre quadrature rule on each subinterval. We prove that the numerical method converges exponentially with respect to both the number of subintervals and the number of Gaussian quadrature points. Some numerical experiments are provided to illustrate the results. [ABSTRACT FROM AUTHOR]

We present a method for computing actions of the exponential-like φ -functions for a Kronecker sum K of d arbitrary matrices A μ . It is based on the approximation of the integral representation of the φ -functions by Gaussian quadrature formulas combined with a scaling and squaring technique. The resulting algorithm, which we call phiks, evaluates the required actions by means of μ -mode products involving exponentials of the small sized matrices A μ , without forming the large sized matrix K itself. phiks, which profits from the highly efficient level 3 BLAS, is designed to compute different φ -functions applied on the same vector or a linear combination of actions of φ -functions applied on different vectors. In addition, thanks to the underlying scaling and squaring techniques, the desired quantities are available simultaneously at suitable time scales. All these features allow the effective usage of phiks in the exponential integration context. In fact, our newly designed method has been tested in popular exponential Runge–Kutta integrators of stiff order from one to four, in comparison with state-of-the-art algorithms for computing actions of φ -functions. The numerical experiments with discretized semilinear evolutionary 2D or 3D advection–diffusion–reaction, Allen–Cahn, and Brusselator equations show the superiority of the proposed μ -mode approach. [ABSTRACT FROM AUTHOR]

The collocation boundary element method, as developed and taught in the traditional books, suffers from severe inconsistencies, partly responsible for the method's lack of clarity and broader applicability (not to mention the uncontrollable proliferation of misleading alternatives that only add to more confusion). This has been recently corrected, as summarized in this review paper, in which we report the proposition of a convergence theorem for the general, just consistent, three-dimensional isoparametric formulation of potential and elasticity problems (https://doi.org/10.1016/j.enganabound.2023.01.017), and also introduce the not interchangeable concepts of nodes and loci, for boundary displacements and tractions in elasticity, respectively, as well as of points, for domain sources. We have implemented, for both two-dimensional potential and elasticity problems, real-variable (https://doi.org/10.1016/j.enganabound.2023.01.015, https://doi.org/10.1016/j.enganabound.2023.03.026) and—still better—complex-variable (https://doi.org/10.1016/j.enganabound.2023.04.024) codes for generally high-order, curved elements, which effortlessly enable evaluations with numerical precision that is only machine-limited and only resorts to the problem's mathematics—plus Gauss–Legendre quadrature of all integrals' regular parts. (Ad hoc mechanical interpretations, "regularizations," interval subdivisions, special quadrature formulas, or variable transformations are neither strictly speaking correct nor generally applicable nor required.) Despite the singular kernels that enter diverse forms of Somigliana's identity (as for evaluating stress results at a domain point), problem-unrelated singularities do not come up in a mathematically consistent formulation and implementation: singularities are just man-made. A few examples of highly challenging topological issues are presented. We show that fracture mechanics problems may be dealt with seamlessly, and we manage to objectively assess a problem's topological consistency, precision, accuracy, and round-off errors related to a given mesh discretization and software-conditioned computational digits. [ABSTRACT FROM AUTHOR]

We consider overdetermined collocation methods and propose a weighted least squares approach to derive a numerical solution. The discrete problem requires the evaluation of the Jacobian of the vector field which, however, appears in a O (h) term, h being the stepsize. We show that, by neglecting this infinitesimal term, the resulting scheme becomes a low-rank Runge–Kutta method. Among the possible choices of the weights distribution, we analyze the one based on the quadrature formula underlying the collocation conditions. A few numerical illustrations are included to better elucidate the potential of the method. [ABSTRACT FROM AUTHOR]

In this article, a generalized log orthogonal functions (GLOFs)‐spectral collocation method to two dimensional weakly singular Volterra integral equations of the second kind is proposed. The mild singularities of the solution at the interval endpoint can be captured by Gauss‐GLOFs quadrature and the shortcoming of the traditional spectral method which cannot well deal with weakly singular Volterra integral equations with limited regular solutions is avoided. A detailed L∞$$ {L}^{\infty } $$ convergence analysis of the numerical solution is carried out. The efficiency of the proposed method is demonstrated by numerical examples. [ABSTRACT FROM AUTHOR]

The numerical integration of definite integrals is essential in fundamental and applied sciences. The accuracy of approximate integral calculations is contingent upon the initial data and specific requirements, leading to the imposition of diverse conditions on the resultant computations. Classical methods for the numerical analysis of definite integrals are known, such as the quadrature formulas of Gregory, Newton-Cotes, Euler, Gauss, Markov, etc. Since the middle of the last century, the theory of constructing optimal formulas for numerical integration based on variational methods began to develop. It should be noted that there are optimal quadrature formulas in the sense of Nikolsky and Sard. In this paper, we study the problem of constructing an optimal quadrature formula in the sense of Sard. When constructing a quadrature formula, the method of φ-functions is used. The error of the formula is estimated from above using the integral of the square of the function from a specific Hilbert space. Next, such a φ function is selected, and the integral of the square in this interval takes the smallest value. The coefficients of the optimal quadrature formula are calculated using the resulting φ function. The optimal quadrature formula in this work is exact on the functions eσx and e-σx, where σ is a nonzero real parameter. [ABSTRACT FROM AUTHOR]

For T n (x) = cos n arccos x , x ∈ [ - 1 , 1 ] , the n-th Chebyshev polynomial of the first kind, we study the quantity τ n , k : = | T n (k) (ω n , k) | T n (k) (1) , 1 ≤ k ≤ n - 2 , where T n (k) is the k-th derivative of T n and ω n , k is the largest zero of T n (k + 1) . Since the absolute values of the local extrema of T n (k) increase monotonically towards the end-points of [ - 1 , 1 ] , the value τ n , k shows how small is the largest critical value of T n (k) relative to its global maximum T n (k) (1) . This is a continuation of our (joint with Alexei Shadrin) paper "On the largest critical value of T n (k) ", SIAM J. Math. Anal.50(3), 2018, 2389–2408, where upper bounds and asymptotic formuae for τ n , k have been obtained on the basis of the Schaeffer–Duffin pointwise upper bound for polynomials with absolute value not exceeding 1 in [ - 1 , 1 ] . We exploit a 1996 result of Knut Petras about the weights of the Gaussian quadrature formulae associated with the ultraspherical weight function w λ (x) = (1 - x 2) λ - 1 / 2 to find an explicit (modulo ω n , k ) formula for τ n , k 2 . This enables us to prove a lower bound and to refine the previously obtained upper bounds for τ n , k . The explicit formula admits also a new derivation of the asymptotic formula approximating τ n , k for n → ∞ . The new approach is simpler, without using deep results about the ordinates of the Bessel function, and allows to better analyze the sharpness of the estimates. [ABSTRACT FROM AUTHOR]

When the time-dependent boundary element method, also termed the pseudo-initial condition method, is employed for solving transient heat conduction problems, the numerical evaluation of domain integrals is necessitated. Consequently, the accurate calculation of the domain integrals is of crucial importance for analyzing transient heat conduction. However, as the time step decreases progressively and approaches zero, the integrand of the domain integrals is close to singular, resulting in large errors when employing standard Gaussian quadrature directly. To solve the problem and further improve the calculation accuracy of the domain integrals, an (α, β) distance transformation is presented. Distance transformation is a simple and efficient method for eliminating near-singularity, typically applied to nearly singular integrals. Firstly, the (α, β) coordinate transformation is introduced. Then, a new distance transformation for the domain integrals is constructed by replacing the shortest distance with the time step. With the new method, the integrand of the domain integrals is substantially smoothed, and the singularity arising from small time steps in the domain integrals is effectively eliminated. Thus, more accurate results can be obtained by the (α, β) distance transformation. Different sizes of time steps, positions of source point, and shapes of integration elements are considered in numerical examples. Comparative studies of the numerical results for the domain integrals using various methods demonstrate that higher accuracy and efficiency are achieved by the proposed method. [ABSTRACT FROM AUTHOR]

Randomized quadratures for integrating functions in Sobolev spaces of order \alpha \ge 1, where the integrability condition is with respect to the Gaussian measure, are considered. In this function space, the optimal rate for the worst-case root-mean-squared error (RMSE) is established. Here, optimality is for a general class of quadratures, in which adaptive non-linear algorithms with a possibly varying number of function evaluations are also allowed. The optimal rate is given by showing matching bounds. First, a lower bound on the worst-case RMSE of O(n^{-\alpha -1/2}) is proven, where n denotes an upper bound on the expected number of function evaluations. It turns out that a suitably randomized trapezoidal rule attains this rate, up to a logarithmic factor. A practical error estimator for this trapezoidal rule is also presented. Numerical results support our theory. [ABSTRACT FROM AUTHOR]

The goal of the paper is to establish quadrature formulas on combinatorial graphs. Three types of quadrature formulas are developed. Quadrature formulas of the first type are obtained through interpolation by variational splines. This set of formulas is exact on spaces of variational splines on graphs. Since bandlimited functions can be obtained as limits of variational splines we obtain quadrature formulas which are approximately exact on spaces of bandlimited functions. Accuracy of this type of quadrature formulas is given in terms of geometry of the set of nodes of splines and in terms of smoothness of functions which is measured by means of the combinatorial Laplace operator. Quadrature formulas of the second type are obtained through point-wise sampling for bandlimited functions and based on existence of certain frames in appropriate subspaces of bandlimited functions. The third type quadrature formulas are based on the average sampling over subgraphs. Our quadrature formulas which are based on sampling are exact on a relevant subspaces of bandlimited functions. The results of the paper have potential applications to problems that arise in data mining. [ABSTRACT FROM AUTHOR]

In this paper we provide a tetrahedra-free algorithm to compute low-cardinality quadrature rules with a given degree of polynomial exactness, positive weights and interior nodes on a polyhedral element with arbitrary shape. The key tools are the notion of Tchakaloff discretization set and the solution of moment-matching equations by Lawson-Hanson iterations for NonNegative Least-Squares. Several numerical tests are presented. The method is implemented in Matlab as open-source software. [ABSTRACT FROM AUTHOR]

In this paper we consider Gaussian type quadrature rules for trigonometric polynomials where an even number of nodes is fixed in advance. For an integrable and nonnegative weight function w on the interval E = [ a , a + 2 π) , a ∈ R , these quadrature rules have the following form ∫ E t (x) w (x) d x = ∑ i = 1 2 k a i t (y i) + ∑ i = 1 2 (n + γ) A i t (x i) , t ∈ T 2 (n + γ) + k − 1 , where the nodes y i ∈ E , i = 1 , 2 , ... , 2 k , are fixed and prescribed in advance, γ ∈ { 0 , 1 / 2 } and T n = { cos ⁡ k x , sin ⁡ k x | k = 0 , 1 , ... , n } , n ∈ N. Also, for γ = 1 / 2 , i.e., for the case of quadrature rules for trigonometric polynomials with odd number of nodes, we consider the optimal sets of quadrature rules in the sense of Borges (see [1,13]) for trigonometric polynomials with even number of fixed nodes. Let n = (n 1 , n 2 , ... , n r) , r ∈ N , be a multi-index and let W = (w 1 , w 2 , ... , w r) be a system of weight functions on the interval E = [ a , a + 2 π) , a ∈ R. The optimal set of quadrature rules with respect to (W , n) , with even number of fixed nodes, have the form ∫ E f (x) w m (x) d x ≈ ∑ i = 1 2 k a m , i f (y i) + ∑ i = 1 2 | n | + 1 A m , i f (x i) , m = 1 , 2 , ... r , where | n | = n 1 + n 2 + ⋯ + n r and the nodes y i ∈ E , i = 1 , 2 , ... , 2 k , are fixed and prescribed in advance. For r = 1 the optimal set of quadrature rules reduces to Gaussian quadrature rule for trigonometric polynomials with odd number of nodes. For all mentioned quadrature rules, in addition to the theoretical results, we will present the method for construction and give appropriate numerical examples. [ABSTRACT FROM AUTHOR]

Quadrature formulas are often constructed to be exact on the space of functions that are easily integrated and that are in some sense similar to the integrand. This motivates us to explore how the known properties of the integrand can be used to improve the accuracy of certain quadrature rules. In the present paper, we propose an l -point Gauss-type quadrature rule G l into which the zeros and poles of the integrand outside the integration interval are incorporated. Formula G l proves to be exact for certain rational functions which have the same zeros and poles as the integrand. It converges, all its nodes are pairwise distinct and belong to the interior of the integration interval, and all its weights are positive. Theoretical results on the remainder term of G l suggest that formula G l is applicable both when the incorporated zeros and poles of the integrand are known exactly, as well as when they are known approximately. To practically and economically estimate the error of G l , some extensions that inherit the l nodes of G l are developed. They are analogous to the Gauss-Kronrod, averaged Gauss, and generalized averaged Gauss quadrature rules. Numerical experiments confirm the accuracy of G l and its extensions. [ABSTRACT FROM AUTHOR]

In this paper, we first construct an integral identity associated with tempered fractional operators. By using this identity, we have found the error bounds for Simpson's second formula, namely Newton–Cotes quadrature formula for differentiable convex functions in the framework of tempered fractional integrals and classical calculus. Furthermore, it is also shown that the newly established inequalities are the extension of comparable inequalities inside the literature. [ABSTRACT FROM AUTHOR]

The principal tool of computation used in this paper is classical Gaussian quadrature on the interval [0,1], which happens to be particularly effective here. Explicit expressions are found for the derivatives of the Ramanujan integral in question, and it is proved that the latter is completely monotone on (0,\infty). As a byproduct, known series expansions for incomplete gamma functions are examined with regard to their convergence properties. The paper also pays attention to another famous integral, the Euler integral — better known as the gamma function — revitalizing a largely neglected part of the function, the part corresponding to negative values of the argument, which plays a prominent role in our work. [ABSTRACT FROM AUTHOR]

Purpose: The paper aims to the size-dependent analysis of functionally graded materials in thermal environment based on the modified couple stress theory using finite element method. Design/methodology/approach: The element formulation is developed within the framework of the penalty unsymmetric finite element method (FEM) in that the C1 continuity requirement is satisfied in weak sense and thus, C0 continuous interpolation enhanced by independent nodal rotation is employed as the test function. Meanwhile, the trial function is designed based on the stress functions and the weighted residual method. Besides, the special Gauss quadrature scheme is employed for integrals of matrices in accordance with the graded variation of the material properties. Findings: The numerical results reveal that in thermal environment, functionally graded materials exhibit better bending performance compared to homogeneous materials, Moreover, the findings also indicate that with an increase in MLSP, the natural frequencies of out-of-plane modes gradually increase, while the natural frequencies of in-plane modes show much less variation, leading to a mode switch phenomenon. Originality/value: The work provides an efficient numerical tool for analyzing and designing the functionally graded structures in thermal environment in practical engineering applications. [ABSTRACT FROM AUTHOR]

We analyse the Gaussian wave packet transform. Based on the Fourier inversion formula and a partition of unity, which is formed by a collection of Gaussian basis functions, a new representation of square-integrable functions is presented. Including a rigorous error analysis, the variants of the wave packet transform are then derived by a discretization of the Fourier integral via different quadrature rules. Based on Gauss–Hermite quadrature, we introduce a new representation of Gaussian wave packets in which the number of basis functions is significantly reduced. Numerical experiments in 1D illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In a simple way, we prove that there is no positive measure d σ on the interval [a, b] such that the corresponding modified anti-Gaussian quadrature formula G ˘ ℓ + 1 (γ) = G ˘ ℓ + 1 (γ ℓ) is also a Chebyshev quadrature formula. In the special case, this result applies to the corresponding anti-Gaussian quadrature formula G ˘ ℓ + 1 . The latter is recently proved in a different way in Notaris [14, Theorem 2.3]. We also show that the only positive and even measure d σ (x) = d σ (- x) on the symmetric interval [ - a , a ] , for which the modified anti-Gaussian quadrature formula G ˘ ℓ + 1 (γ) = G ˘ ℓ + 1 (γ ℓ) has the form ∫ - a a f (x) d σ (x) = μ 0 2 [ f (- a) + f (a) ] + R 2 (γ ℓ) (f) for ℓ = 1 and ∫ - a a f (x) d σ (x) = w 1 f (- a) + w ∑ k = 1 ℓ - 1 f (t k) + w 1 f (a) + R ℓ + 1 (γ ℓ) (f) for all ℓ ≥ 2 , is the measure d σ (x) = (a 2 - x 2) - 1 / 2 d x . This result is a generalization of the one for the anti-Gaussian quadrature formula G ˘ ℓ + 1 in Notaris [14, Theorem 3.1] to the modified anti-Gaussian quadrature formulas G ˘ ℓ + 1 (γ) = G ˘ ℓ + 1 (γ ℓ) . [ABSTRACT FROM AUTHOR]

Highly oscillatory integrals of composite type arise in electronic engineering and their calculations are a challenging problem. In this paper, we propose two Gaussian quadrature rules for computing such integrals. The first one is constructed based on the classical theory of orthogonal polynomials and its nodes and weights can be computed efficiently by using tools of numerical linear algebra. An interesting connection between the quadrature nodes and the Legendre points is proved and it is shown that the rate of convergence of this rule depends solely on the regularity of the non-oscillatory part of the integrand. The second one is constructed with respect to a sign-changing function and the classical theory of Gaussian quadrature cannot be used anymore. We explore theoretical properties of this Gaussian quadrature, including the trajectories of the quadrature nodes and the convergence rate of these nodes to the endpoints of the integration interval, and prove its asymptotic error estimate under suitable hypotheses. Numerical experiments are presented to demonstrate the performance of the proposed methods. [ABSTRACT FROM AUTHOR]

Solutions to problems arising from much scientific and applied research conducted at the world level lead to integral and differential equations. They are approximately solved, mainly using quadrature, cubature, and difference formulas. Therefore, in the current work, we consider a discrete analogue of the differential operator 1 − 1 2 π 2 d 2 d x 2 m in the Hilbert space H 2 μ R , called D m β . We modify the Sobolev algorithm to construct optimal quadrature formulas using a discrete operator. We provide a weighted optimal quadrature formula, using this algorithm for the case where m = 1 . Finally, we construct an optimal quadrature formula in the Hilbert space H 2 μ R for the weight functions p x = 1 and p x = e 2 π i ω x when m = 1 . [ABSTRACT FROM AUTHOR]

Polynomial interpolation is an important component of many computational problems. In several of these computational problems, failure to preserve positivity when using polynomials to approximate or map data values between meshes can lead to negative unphysical quantities. Currently, most polynomial-based methods for enforcing positivity are based on splines and polynomial rescaling. The spline-based approaches build interpolants that are positive over the intervals in which they are defined and may require solving a minimization problem and/or system of equations. The linear polynomial rescaling methods allow for high-degree polynomials but enforce positivity only at limited locations (e.g., quadrature nodes). This work introduces open-source software (HiPPIS) for high-order data-bounded interpolation (DBI) and positivity-preserving interpolation (PPI) that addresses the limitations of both the spline and polynomial rescaling methods. HiPPIS is suitable for approximating and mapping physical quantities such as mass, density, and concentration between meshes while preserving positivity. This work provides Fortran and Matlab implementations of the DBI and PPI methods, presents an analysis of the mapping error in the context of PDEs, and uses several 1D and 2D numerical examples to demonstrate the benefits and limitations of HiPPIS. [ABSTRACT FROM AUTHOR]

In this paper, the numerical algorithms for calculating the values of the left- and right-sided Riemann–Liouville fractional integrals and the Riesz fractional integral using spline interpolation techniques are derived. The linear, quadratic and three variants of cubic splines are taken into account. The estimation of errors using analytical methods are derived. We show four examples of numerical evaluation of the mentioned fractional integrals and determine the experimental rate of convergence for each derived algorithm. The high-precision calculations are executed using the 128-bit floating-point numbers and arithmetic routines. [ABSTRACT FROM AUTHOR]

This study mathematically examines chemical and biomaterial models by employing the finite element method. Unshaped biomaterials' complex structures have been numerically analyzed using Gaussian quadrature rules. It has been analyzed for commercial benefits of chemical engineering and biomaterials as well as biorefinery fields. For the computational work, the ellipsoid has been taken as a model, and it has been transformed by subdividing it into six tetrahedral elements with one curved face. Each curved tetrahedral element is considered a quadratic and cubic tetrahedral element and transformed into standard tetrahedral elements with straight faces. Each standard tetrahedral element is further decomposed into four hexahedral elements. Numerical tests are presented that verify the derived transformations and the quadrature rules. Convergence studies are performed for the integration of rational, weakly singular, and trigonometric test functions over an ellipsoid by using Gaussian quadrature rules and compared with the generalized Gaussian quadrature rules. The new transformations are derived to compute numerical integration over curved tetrahedral elements for all tests, and it has been observed that the integral outcomes converge to accurate values with lower computation duration. [ABSTRACT FROM AUTHOR]

To numerically simulate the acoustic propagation in fluids, a third-order compact finite volume method (FVM) that is suitable for three-dimensional (3D) hybrid unstructured grids is developed. It is used to solve the linear Euler equations (LEEs). Furthermore, the corresponding linear lattice Boltzmann flux solver (LBFS) is constructed. The conservative form of LEEs is discretized according to the third-order Gaussian quadrature rules. Based on the stencils only involving the neighboring cells of the control volume, high-order polynomials are constructed. By ensuring that the variables and their derivatives are conserved in the stencil, the unknown coefficients are determined using the least-square method. To satisfy the moment relations, the distribution function expression is determined by solving the equation system. Finally, the mesoscopic flux calculation scheme for LEEs is constructed based on the derived lattice model. The proposed high-order FVM is suitable for 3D hybrid unstructured grids, which makes it highly adaptable. It makes the selection of stencils and disposal of ghost cells simple due to its compact characteristics. Moreover, the proposed FVM extends the application of LBFS from traditional fluid equations to 3D LEEs. Therefore, the proposed flux solver has the advantages of Boltzmann-based flux schemes. The proposed FVM is used to simulate 3D acoustic propagations and is validated through several numerical cases. [ABSTRACT FROM AUTHOR]

The paper presents a technique of motion planning for autonomous vehicles (AV) based on simultaneous trajectory and speed optimization. The method includes representing the trajectory by a finite element (FE), determining trajectory parameters in Frenet coordinates, composing a model of vehicle kinematics, defining optimization criteria and a cost function, forming a set of constraints, and adapting the Gaussian N-point scheme for quadrature numerical integration. The study also defines a set of minimum optimization parameters sufficient for making motion predictions with smooth functions of the trajectory and speed. For this, piecewise functions with three degrees of freedom (DOF) in FE's nodes are implemented. Therefore, the high differentiability of the trajectory and speed functions is ensured to obtain motion criteria such as linear and angular speeds, acceleration, and jerks used in the cost function and constraints. To form the AV roadway position, the Frenet coordinate system and two variable parameters are used: the reference path length and the lateral displacement perpendicular to reference line's tangent. The trajectory shape, then, depends only on the final position of the AV's mass center and the final reference's curvature. The method uses geometric, kinematic, dynamic, and physical constraints, some of which are related to hard restrictions and some to soft restrictions. The planning technique involves parallel forecasting for several variants of the AV maneuver followed by selecting the one corresponding to a specified criterion. The sequential quadratic programming (SQP) technique is used to find the optimal solution. Graphs of trajectories, speeds, accelerations, jerks, and other parameters are presented based on the simulation results. Finally, the efficiency, rapidity, and prognosis quality are evaluated. [ABSTRACT FROM AUTHOR]

In this work, our primary focus is on the numerical computation of highly oscillatory integrals involving the Airy function. Specifically, we address integrals of the form ∫ 0 b x α f (x) Ai (− ω x) d x over a finite or semi-infinite interval, where the integrand exhibits rapid oscillations when ω ≫ 1 . The inherent high oscillation and algebraic singularity of the integrand make traditional quadrature rules impractical. In view of this, we strategically partition the interval into two segments: [ 0 , 1 ] and [ 1 , b ] . For integrals over the interval [ 0 , 1 ] , we introduce a Filon-type method based on a two-point Taylor expansion. In contrast, for integrals over [ 1 , b ] , we transform the Airy function into the first kind of Bessel function. By applying Cauchy's integration theorem, the integral is then reformulated into several non-oscillatory and exponentially decaying integrals over [ 0 , + ∞) , which can be accurately approximated by the generalized Gaussian quadrature rule. The proposed methods are accompanied by rigorous error analyses to establish their reliability. Finally, we present a series of numerical examples that not only validate the theoretical results but also showcase the accuracy and efficacy of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a new boundary-conforming adaptive method for the numerical integration of trimmed elements is presented. The locations and weights of new integration points are determined based on special mapping formulations. The prerequisite of this technique is to describe the trimming curves/surfaces by parametric Bezier curves/surfaces within the parent element domain. The fitting error is under control, therefore the number of quadrature points for exact integration can be adjusted automatically based on the complexity of trimming curve and the corresponding integrand. The proposed method can be easily implemented into fictitious domain approaches. Different integration tasks as well as structural examples reveal that the proposed method delivers accurate and robust solutions for a wide variety of 2D/3D geometries with a rather low number of integration points. [ABSTRACT FROM AUTHOR]

We show that the eigenpolytopes of graphs are universal in the sense that every polytope, up to affine equivalence, appears as the eigenpolytope of some positively weighted graph. We next extend the theory of graphical designs, which are quadrature rules for graphs, to positively weighted graphs. Through Gale duality for polytopes, we show a bijection between graphical designs and the faces of eigenpolytopes. This bijection proves the existence of graphical designs with positive quadrature weights and upper bounds the size of a minimal graphical design. Connecting this bijection with the universality of eigenpolytopes, we establish three complexity results: It is strongly NP-complete to determine if there is a graphical design smaller than the mentioned upper bound, it is NP-hard to find a smallest graphical design, and it is \#P-complete to count the number of minimal graphical designs. [ABSTRACT FROM AUTHOR]

We construct optimal quadrature formulas for numerical integration of the right Riemann–Liouville integrals in the Hilbert space W 2 m , m - 1 t , 1 . We obtain a system of linear equations for the coefficients of the optimal quadrature formula and establish The existence and uniqueness of its solution. Explicit expressions of the optimal coefficients are obtained. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of the spectrum of an integral operator similar to an integral operator with a logarithmic kernel depending on the sum of arguments. By a simple change of variables, the corresponding equation is reduced to an integral equation of convolution type defined on a finite interval (as is well known, such equations in the general case cannot be solved by quadratures). Next, using the Fourier transform, the equation is reduced to a conjugation problem for analytic functions and then to an infinite system of linear algebraic equations, the isolation of the main terms in which allows deriving a relation that determines the spectrum of the original problem. [ABSTRACT FROM AUTHOR]

Using the formulation of the electromagnetic Green's function of a perfectly conducting cone in terms of analytically continued angular momentum, we compute the Casimir–Polder interaction energy of a cone with a polarizable particle. We introduce this formalism by first reviewing the analogous approach for a perfectly conducting wedge, and then demonstrate the calculation through numerical evaluation of the resulting integrals. [ABSTRACT FROM AUTHOR]