Abstract. In this paper, we present a theory for bounding the minimum eigenvalues, maximum eigenvalues, and condition numbers of stiffness matrices arising from the p-version of finite element analysis. Bounds are derived for the eigenvalues and the condition numbers, which are valid for stiffness matrices based on a set of general basis functions that can be used in the p-version. For a set of hierarchical basis functions satisfying the usual local support condition that has been popularly used in the p-version, explicit bounds are derived for the minimum eigenvalues, maximum eigenvalues, and condition numbers of stiffness matrices. We prove that the condition numbers of the stiffness matrices grow like p 4(d−1) , where d is the number of dimensions. Our results disprove a conjecture of Olsen and Douglas in which the authors assert that "regardless of the choice of basis, the condition numbers grow like p 4d or faster". Numerical results are also presented which verify that our theoretical bounds are correct.

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As an important tool for monitoring the marine environment, safeguarding maritime rights and interests and building a smart ocean, underwater equipment has developed rapidly in recent years. Due to the problems of seawater corrosion, excessive deep-sea static pressure and noise interference in the marine environment and economy, the applicability of manufacturing materials must be considered at the beginning of the design of underwater equipment. Piezoelectric metamaterial is widely used in underwater equipment instead of traditional materials because the traditional materials can not meet the application requirements. In this paper, according to the application range of piezoelectric metamaterials in underwater equipment, the current application of piezoelectric metamaterials is reviewed from the aspects of sound insulation and energy conversion. On this basis, the future development prospect of piezoelectric metamaterials in underwater equipment is introduced.

The fractional-order differential operator describes history dependence and global correlation. In this paper, we use this trait to describe the viscoelastic characteristics of the solid skeleton of a viscoelastic two-phasic porous material. Combining Kjartansson constant Q fractional order theory with the BISQ theory, a new BISQ model is proposed to simulate elastic wave propagation in a viscoelastic two-phasic porous material. The corresponding time-domain wave propagation equations are derived, and then the elastic waves are numerically simulated in different cases. The integer-order derivatives are discretised using higher-order staggered-grid finite differences, and the fractional-order time derivatives are discretised using short-time memory central differences. Numerical simulations and analysis of the wave field characterisation in different phase boundaries, different quality factor groups, and multilayered materials containing buried bodies are carried out. The simulation results show that it is feasible to combine the constant Q fractional-order derivative theory with the BISQ theory to simulate elastic waves in viscoelastic two-phasic porous materials. The combination can better describe the viscoelastic characteristics of the viscoelastic two-phasic porous materials, which is of great significance for further understanding the propagation mechanism of elastic waves in viscoelastic two-phasic porous materials and viscoelastic two-phasic porous materials containing buried bodies. This paper provides a theoretical forward simulation for fine inversion and reconstruction of layer information and buried body structure in viscoelastic two-phasic porous materials.

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