Ravana anugrahamurthi is none other Lord Shiva himself. This name says that he showered his grace on the demon king Ravana in the epic Ramayana. Ravana is one of the profound devotees of Lord Shiva.
It is said that demon Ravana is the foremost devotee to Lord Shiva ever in the history. He is the only one who impressed Lord Shiva more than a couple of times in his life time with his intensified penance and pure dedication towards Lord Shiva. Even the Gods, Sages, Priests and other divine Personalities did not have the power of concentration Ravana had.
It is believed that it is Ravana who got the City Lanka from the Lord Shiva. Ravana was a born Brahmin and he conducted the opening ceremony of the city Lanka, which has been designed and built by one of the celestial architects for the couple Shiva and Parvati. The moment the city has been prepared to hand over to Lord Shiva, Ravana had requested Shiva to fulfill his wish. Ravana had asked the city as the compensation for performing the opening ceremony for the city.
Lord Shiva, who is at most compassionate towards his devotees, had granted the city to Ravana without a second thought. Though Goddess Parvati was against this, she did not intervene since it the nature of Shiva to shower his grace on his devotees.
In the other situation, it was evident that Ravana had taken out his nerves out of the body to pray Lordd Shiva by playing Veena using them. Delighted Lord Shiva granted him whatever has been desired by Ravana.