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The purpose of this paper is to develop an easy-to-implement and accurate fast boundary element method (BEM) for solving large-scale elastodynamic problems in frequency and time domains.

A newly developed kernel-independent fast multipole method (KIFMM) is applied to accelerating the evaluation of displacements, strains and stresses in frequency domain elastodynamic BEM analysis, in which the far-field interactions are evaluated efficiently utilizing equivalent densities and check potentials. Although there are six boundary integrals with unique kernel functions, by using the elastic theory, the authors managed to accelerate these six boundary integrals by KIFMM with the same kind of equivalent densities and check potentials. The boundary integral equations are discretized by Nyström method with curved quadratic elements. The method is further used to conduct the time-domain analysis by using the frequency-domain approach.

Numerical results show that by the fast BEM, high accuracy can be achieved and the computational complexity is brought down to linear. The performance of the present method is further demonstrated by large-scale simulations with more than two millions of unknowns in the frequency domain and one million of unknowns in the time domain. Besides, the method is applied to the topological derivatives for solving elastodynamic inverse problems.

An efficient KIFMM is implemented in the acceleration of the elastodynamic BEM. Combining with the Nyström discretization based on quadratic elements and the frequency-domain approach, an accurate and highly efficient fast BEM is achieved for large-scale elastodynamic frequency domain analysis and time-domain analysis.

Based on the asymptotic solution for predicted natural frequencies of a two‐dimensional elastodynamic problem from the finite element analysis, presents the concept of the asymptotic error, which is an approximate error but tends to the exact error when the characteristic length of elements approaches zero, and a practical error estimator. The present practical error estimator contains two criteria: one is the error estimator criterion, the other the finite element mesh design criterion. Using this practical error estimator, not only can the accuracy of a finite element solution for natural frequencies of a two‐dimensional elastodynamic problem be directly evaluated without any further finite element calculation, but also a new target finite element mesh for the desired accuracy of solution can be immediately designed from the relevant information of an original finite element solution. Generally, for the purpose of designing a new target finite element mesh, this original finite element solution is obtainable from a very coarse mesh of a few elements and usually does not satisfy the accuracy requirement. Since the new target finite element mesh could result in a finite element solution with a desire accuracy, the finite element solution so obtained can be used for a structural design in engineering practice. The related numerical results from vibration problems of three representative plates of different shapes under plane stress conditions have demonstrated the correctness and applicability of the present practical error estimator.

The 2D frequency domain boundary integral equation is solved by the boundary spectral strip method. Using an expansion for frequency domain elastodynamics kernel we reduce its singularity and present analytical solutions for the required integrals in the singular case when the integration path is a straight line. The method is illustrated by two different problems, both over a range of excitation frequencies. The first problem is a rectangular bar under a longitudinal excitation, which has an analytical solution. The other problem is a trapezoidal dam loaded by a transverse excitation at its base. The solution for the second problem is compared with a finite elements model. The results obtained from these tests show a good agreement between the results of the boundary strip method and analytical or finite elements results.

The elastodynamic response of two coplanar finite cracks in an infinite orthotropic strip due to longitudinal waves incident normally on the cracks has been investigated in this paper. Fourier transforms are used to reduce the mixed boundary value problem to Cauchy‐type singular integral equation, which is solved numerically. The stress intensity factors at the edges of the cracks have been derived and plotted against dimensionless frequency for different geometry parameters. Also normal stress out side the cracks has been calculated numerically and plotted for different parameters and different materials.

The Convolution Quadrature Method (CQM) has been widely applied to solve transient elastodynamic problems because of its stability and generality. However, the CQM suffers from the problems of huge memory requirement in case of direct implementation in time domain or CPU time in case of its reformulation in Laplace domain. The purpose of this paper is to combine the CQM with the pseudo-initial condition method (PICM) to achieve a good balance between memory requirement and CPU time.

The combined methods first subdivide the whole analysis into a few sub-analyses, which is dealt with the PICM, namely, the results obtained by previous sub-analysis are used as the initial conditions for the next sub-analysis. In each sub-analysis, the time interval is further discretized into a number of sub-steps and dealt with the CQM. For non-zero initial conditions, the pseudo-force method is used to transform them into equivalent body forces. The boundary face method is employed in the numerical implementation. Three examples are analyzed. Results are compared with analytical solutions or FEM results and the results of reformulated CQM.

Results demonstrate that the computation time and the storage requirement can be reduced significantly as compared to the CQM, by using the combined approach.

The combined methods can be successfully applied to the problems of long-time dynamic response, which requires a large amount of computer memory when CQM is applied, while preserving the CQM stability. If the number of time steps is high, then the accuracy of the proposed approach can be deteriorated because of the pseudo-force method.

The purpose of this paper is to calculate near field and far field scattering of SH waves by multiple multilayered anisotropic circular inclusions using parallel volume integral equation method (PVIEM) quantitatively.

The PVIEM is applied for the analysis of elastic wave scattering problems in an unbounded solid containing multiple multilayered anisotropic circular inclusions. It should be noted that this numerical method does not require the use of the Green’s function for the inclusion – only the Green’s function for the unbounded isotropic matrix is needed. This method can also be applied to solve general elastodynamic problems involving inhomogeneous and/or anisotropic inclusions whose shape and number are arbitrary.

A detailed analysis of the SH wave scattering problem is presented for multiple multilayered orthotropic circular inclusions. Numerical results are presented for the displacement fields at the interfaces and the far field scattering patterns for square and hexagonal packing arrays of multilayered circular inclusions in a broad frequency range of practical interest.

To the best of the authors’ knowledge, the solution for scattering of SH waves by multiple multilayered anisotropic circular inclusions in an unbounded isotropic matrix is not currently available in the literature. However, in this paper, calculation of displacements on interfaces and far field scattering patterns of multiple multilayered anisotropic circular inclusions using PVIEM as a pioneer of numerical modeling enables us to investigate the effects of single/multiple scattering, fiber packing type, fiber volume fraction, single/multiple layer(s), the multilayer’s geometry, isotropy/anisotropy and softness/hardness.

Direct and indirect time marching boundary element methods often become numerically unstable. Evidence of, and reasons for, these instabilities is provided in this paper. Two new time stepping schemes are presented, both of which are more stable than the existing standard schemes available. In particular, we introduce the Half‐step scheme, which is more accurate and far more stable than existing methods. This scheme, which is demonstrated on a simple crack problem for the displacement discontinuity method, can also be introduced into the direct boundary element method. Implementation of the Half‐step scheme into existing boundary element codes will allow researchers to attack more challenging problems than before.

The purpose of this paper is to improve the effectiveness of ordinary reduction methods performance, in nonlinear dynamic analysis. In this paper, the error vector due to linear and nonlinear dynamic analysis in generalized subspaces is extracted, and is decomposed into two independent components, namely outside and inside components. Based on the inside error component, a new iterative reduction method, one‐dimensional generalized subspace procedure (ODGS), is proposed where an innovative criterion is defined for updating the base vectors necessary for stiffness changes in nonlinear dynamic analysis. In this study, the performance of ODGS for linear and nonlinear analysis of elastodynamic systems including non‐proportional damping based on the Ritz generalized subspace has been proposed. Numerical examples show the competency of the proposed method in both economy and exactness. Time saving gained from the ODGS method could be recompensed to get much more accurate results consuming the same CPU time. This iterative method is more effective than the ordinary reduction methods. Since the method is directly derived from the discrete model based on the finite element method (FEM), the complexity of the structure does not affect directly the effectiveness of ODGS. Therefore, whenever the FEM is effectively capable to represent the topology of the structure, the ODGS results will also represent the system response properly. Same as any other reduction methods, accuracy of this iterative reduction method is directly related to the number of selected Ritz vectors, according to convergence criterion.

The purpose of this paper is to present a novel strategy used for acceleration of free-vibration analysis, in which the hierarchical matrices structure and Compute Unified Device Architecture (CUDA) platform is applied to improve the performance of the traditional dual reciprocity boundary element method (DRBEM).

The DRBEM is applied in forming integral equation to reduce complexity. In the procedure of optimization computation, ℋ-Matrices are introduced by applying adaptive cross-approximation method. At the same time, this paper proposes a high-efficiency parallel algorithm using CUDA and the counterpart of the serial effective algorithm in ℋ-Matrices for inverse arithmetic operation.

The analysis for free-vibration could achieve impressive time and space efficiency by introducing hierarchical matrices technique. Although the serial algorithm based on ℋ-Matrices could obtain fair performance for complex inversion operation, the CUDA parallel algorithm would further double the efficiency. Without much loss in accuracy according to the examination of the numerical example, the relative error appeared in approximation process can be fixed by increasing degrees of freedoms or introducing certain amount of internal points.

The paper proposes a novel effective strategy to improve computational efficiency and decrease memory consumption of free-vibration problems. ℋ-Matrices structure and parallel operation based on CUDA are introduced in traditional DRBEM.

This paper aims to present the optimal design procedure of a symmetrical 2-DOF parallel planar robot with flexible joints by considering several performance criteria based on the workspace size, dynamic dexterity and energy of the control.

Consequently, the optimal design consists in determining the dimensional parameters to maximize the size of the workspace, maximize the dynamic dexterity and minimize the energy of the control action. The design criteria are derived from the kinematics, dynamics, elastodynamics and the position control law of the robot. The analysis of the design criteria is performed by means of the design space and atlases.

Finally, the multi-objective design optimization derived from the optimal design procedure is solved by using multi-objective genetic algorithms, and the results are analyzed to assess the validity of the proposed approach.

An alternative approach to the design of a planar parallel robot with flexible joints that permits determining the structural parameters by considering kinematic, dynamic and control operational performance.