## 1 Introduction  
Therapeutic ultrasound has been wildly applied to treat patients with muscle spasm/pain or musculoskeletal injuries and to promote soft-tissue healing. During thermal therapy, the nominal intensity of ultrasound is 0.8-2.0 \(W\)/cm\({}^{2}\) and the ultrasound transducer is moved manually at a rate of 2-4 cm/s with a linear or circular displacement pattern [1; 2; 3; 4]. It is generally recommended to heat soft tissues below 45 \({}^{\circ}\)C for at least five minutes. When the temperature of soft tissues reaches beyond 45 \({}^{\circ}\)C, it is very likely to cause protein denaturation and tissue damage, which is not in line with the expected treatment goals [5]. Clinically, surgical implants can be used to fill tissues, fix prostheses, or protect bones in humans. Patients with implants might suffer joint contractors [6; 7; 8], and heat treatments are believed to help reduce the symptoms of associated complications [9; 10]. Among the many available thermal therapy devices, ultrasound can heat deep tissues to reduce muscle spasm and pain without affecting the surface structure. Although the biophysical effects of ultrasound on tissues have been studied for many years, most studies have excluded the presence of surgical implants during ultrasound diathermy. Only a few studies have considered the presence of implants during ultrasound diathermy and discussed the effect of implants on the temperature field in the tissues near the implant [11; 12; 13; 14; 15; 16; 17; 18]. However, the findings of these studies are contradictory. Early clinical staff believed that ultrasound waves could be reflected by metal implants, resulting in overheating at the reflection focal point in the tissue [14]. However, in vitro experimental results have indicated that temperature increases in front of metal plate implants are smaller than in front of bones. Due to the high thermal conductivity of metals, the presence of metallic objects causes significant heat loss and does not lead to a clear temperature rise in front of the metal objects [11; 12; 13; 14]. By contrast, [PERSON] et al. [16] found that the intramuscular temperature was significantly higher under a 3 MHz frequency ultrasound operation in the presence of the metal plate from in vitro experiments. [PERSON] et al. [18] discussed a clinical case that reported a direct association between ultrasound therapy and fatigue fracture of a femoral stem with a metallic material. These investigators warned that continuous ultrasound diathermy with metallic implants may lead to undesirable effects. According to previous literature review, it is found that some unknown factors remain regarding the effect of implants on ultrasonic thermal therapies. The temperature distributions in tissues are affected by not only the ultrasound operating frequency and intensity but also the inclusion geometry, position, and material during ultrasound diathermy. Due to the presence of curved inclusions, both reflected and diffracted ultrasound waves simultaneously exist in front of the implant. Such resultant wave fields may disturb the temperature distribution, differently than those for a flat inclusion. However, only few research studies [19; 20; 21] have investigated the interfacial effect in a composite system, and there still lacks a systematic study to discuss the influences of non-flat inclusions on ultrasonic diathermy in a composite system of different materials. In the present study, the actual tissue-bone system is idealized as a two-dimensional (2-D) bilayer composite system for simplicity. The top layer is composed of a soft tissue and the bottom of the tissue is attached to a bone with a convex surface. Furthermore, a circular inclusion, which can be considered as a cross-section of a cylindrical implant, such as a bone nail, surgical implant, or vessel, is embedded in the tissue. The composite system is heated by an ultrasound probe on the surface of the tissue.